System and method for net load-based inference of  operational specifications of a photovoltaic power generation system with the aid of a digital computer

ABSTRACT

A system and method for net load-based inference of operational specifications of a photovoltaic power generation system with the aid of a digital computer are provided. Photovoltaic plant configuration specifications can be accurately inferred with net load data and measured solar resource data. Power generation data is simulated for a range of hypothetical photovoltaic system configurations based on a normalized solar power simulation model. Net load data is estimated based on one or more component loads. The set of key parameters corresponding to the net load estimate that minimizes total squared error represents the inferred specifications of the photovoltaic plant configuration.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation of U.S. patent applicationSer. No. 14/224,018, filed Mar. 24, 2014, pending; which is acontinuation-in-part of U.S. Pat. No. 8,682,585, issued Mar. 25, 2014;which is a continuation of U.S. Pat. No. 8,437,959, issued May 7, 2013,pending; which is a continuation of U.S. Pat. No. 8,335,649, issued Dec.18, 2012, pending; which is a continuation of U.S. Pat. No. 8,165,812,issued Apr. 24, 2012, the priority dates of which are claimed and thedisclosures of which are incorporated by reference.

This invention was made with State of California support under AgreementNumber 722. The California Public Utilities Commission of the State ofCalifornia has certain rights to this invention.

FIELD

This application relates in general to photovoltaic power generationfleet planning and operation and, in particular, to a system and methodfor net load-based inference of operational specifications of aphotovoltaic power generation system with the aid of a digital computer.

BACKGROUND

The manufacture and usage of photovoltaic systems has advancedsignificantly in recent years due to a continually growing demand forrenewable energy resources. The cost per watt of electricity generatedby photovoltaic systems has decreased dramatically, especially whencombined with government incentives offered to encourage photovoltaicpower generation. Photovoltaic systems are widely applicable asstandalone off-grid power systems, sources of supplemental electricity,such as for use in a building or house, and as power grid-connectedsystems. Typically, when integrated into a power grid, photovoltaicsystems are collectively operated as a fleet, although the individualsystems in the fleet may be deployed at different physical locationswithin a geographic region.

Grid connection of photovoltaic power generation fleets is a fairlyrecent development. In the United States, the Energy Policy Act of 1992deregulated power utilities and mandated the opening of access to powergrids to outsiders, including independent power providers, electricityretailers, integrated energy companies, and Independent System Operators(ISOs) and Regional Transmission Organizations (RTOs). A power grid isan electricity generation, transmission, and distribution infrastructurethat delivers electricity from supplies to consumers. As electricity isconsumed almost immediately upon production, power generation andconsumption must be balanced across the entire power grid. A large powerfailure in one part of the grid could cause electrical current toreroute from remaining power generators over transmission lines ofinsufficient capacity, which creates the possibility of cascadingfailures and widespread power outages.

As a result, planners and operators of power grids need to be able toaccurately gauge both on-going and forecasted power generation andconsumption. Photovoltaic fleets participating as part of a power gridare expected to exhibit predictable power generation behaviors. Powerproduction data is needed at all levels of a power grid to which aphotovoltaic fleet is connected. Accurate power production data isparticularly crucial when a photovoltaic fleet makes a significantcontribution to a power grid's overall energy mix. At the individualphotovoltaic plant level, power production forecasting first involvesobtaining a prediction of solar irradiance, which can be derived fromground-based measurements, satellite imagery, numerical weatherprediction models, or other sources. The predicted solar irradiance dataand each photovoltaic plant's system configuration is then combined witha photovoltaic simulation model, which generates a forecast ofindividual plant power output production. The individual photovoltaicplant forecasts can then be combined into a photovoltaic powergeneration fleet forecast, such as described in commonly-assigned U.S.Pat. Nos. 8,165,811; 8,165,812; 8,165,813, all issued to Hoff on Apr.24, 2012; U.S. Pat. Nos. 8,326,535; 8,326,536, issued to Hoff on Dec. 4,2012; and U.S. Pat. No. 8,335,649, issued to Hoff on Dec. 18, 2012, thedisclosures of which are incorporated by reference.

A grid-connected photovoltaic fleet can be operationally dispersed overa neighborhood, utility region, or several states, and its constituentphotovoltaic systems (or plants) may be concentrated together or spreadout. Regardless, the aggregate grid power contribution of a photovoltaicfleet is determined as a function of the individual power contributionsof its constituent photovoltaic plants, which, in turn, may havedifferent system configurations and power capacities. Photovoltaicsystem configurations are critical to forecasting plant power output.Inaccuracies in the assumed specifications of photovoltaic systemconfigurations directly translate to inaccuracies in their power outputforecasts. Individual photovoltaic system configurations may vary basedon power rating and electrical characteristics and by their operationalfeatures, such as tracking mode (fixed, single-axis tracking, dual-axistracking), azimuth angle, tilt angle, row-to-row spacing, trackingrotation limit, and shading or other physical obstructions.

Photovoltaic system power output is particularly sensitive to shadingdue to cloud cover, and a photovoltaic array with only a small portioncovered in shade can suffer a dramatic decrease in power output. For asingle photovoltaic system, power capacity is measured by the maximumpower output determined under standard test conditions and is expressedin units of Watt peak (Wp). However, at any given time, the actual powercould vary from the rated system power capacity depending upongeographic location, time of day, weather conditions, and other factors.Moreover, photovoltaic fleets with individual systems scattered over alarge geographical area are subject to different location-specific cloudconditions with a consequential effect on aggregate power output.

Consequently, photovoltaic fleets operating under cloudy conditions canexhibit variable and unpredictable performance. Conventionally, fleetvariability is determined by collecting and feeding direct powermeasurements from individual photovoltaic systems or equivalentindirectly derived power measurements into a centralized controlcomputer or similar arrangement. To be of optimal usefulness, the directpower measurement data must be collected in near real time at finegrained time intervals to enable a high resolution time series of poweroutput to be created. However, the practicality of such an approachdiminishes as the number of systems, variations in systemconfigurations, and geographic dispersion of the photovoltaic fleetgrow. Moreover, the costs and feasibility of providing remote powermeasurement data can make high speed data collection and analysisinsurmountable due to the bandwidth needed to transmit and the storagespace needed to contain collected measurements, and the processingresources needed to scale quantitative power measurement analysisupwards as the fleet size grows.

For instance, one direct approach to obtaining high speed time seriespower production data from a fleet of existing photovoltaic systems isto install physical meters on every photovoltaic system, record theelectrical power output at a desired time interval, such as every 10seconds, and sum the recorded output across all photovoltaic systems inthe fleet at each time interval. The totalized power data from thephotovoltaic fleet could then be used to calculate the time-averagedfleet power, variance of fleet power, and similar values for the rate ofchange of fleet power. An equivalent direct approach to obtaining highspeed time series power production data for a future photovoltaic fleetor an existing photovoltaic fleet with incomplete metering and telemetryis to collect solar irradiance data from a dense network of weathermonitoring stations covering all anticipated locations of interest atthe desired time interval, use a photovoltaic performance model tosimulate the high speed time series output data for each photovoltaicsystem individually, and then sum the results at each time interval.

With either direct approach to obtaining high speed time series powerproduction data, several difficulties arise. First, in terms of physicalplant, calibrating, installing, operating, and maintaining meters andweather stations is expensive and detracts from cost savings otherwiseafforded through a renewable energy source. Similarly, collecting,validating, transmitting, and storing high speed data for everyphotovoltaic system or location requires collateral data communicationsand processing infrastructure, again at possibly significant expense.Moreover, data loss occurs whenever instrumentation or datacommunications do not operate reliably.

Second, in terms of inherent limitations, both direct approaches toobtaining high speed time series power production data only work fortimes, locations, and photovoltaic system configurations when and wheremeters are pre-installed; thus, high speed time series power productiondata is unavailable for all other locations, time periods, andphotovoltaic system configurations. Both direct approaches also cannotbe used to directly forecast future photovoltaic system performancesince meters must be physically present at the time and location ofinterest. Fundamentally, data also must be recorded at the timeresolution that corresponds to the desired output time resolution. Whilelow time-resolution results can be calculated from high resolution data,the opposite calculation is not possible. For example, photovoltaicfleet behavior with a 10-second resolution cannot be determined fromdata collected by existing utility meters that collect the data with a15-minute resolution.

The few solar data networks that exist in the United States, such as theARM network, described in G. M. Stokes et al., “The atmosphericradiation measurement (ARM) program: programmatic background and designof the cloud and radiation test bed,” Bulletin of Am. Meteor. Soc., Vol.75, pp. 1201-1221 (1994), the disclosure of which is incorporated byreference, and the SURFRAD network, do not have high density networks(the closest pair of stations in the ARM network are 50 km apart) norhave they been collecting data at a fast rate (the fastest rate is 20seconds in the ARM network and one minute in the SURFRAD network). Thelimitations of the direct measurement approaches have promptedresearchers to evaluate other alternatives. Researchers have installeddense networks of solar monitoring devices in a few limited locations,such as described in S. Kuszamaul et al., “Lanai High-Density IrradianceSensor Network for Characterizing Solar Resource Variability of MW-ScalePV System.” 35^(th) Photovoltaic Specialists Conf., Honolulu, Hi. (Jun.20-25, 2010), and R. George, “Estimating Ramp Rates for Large PV SystemsUsing a Dense Array of Measured Solar Radiation Data,” Am. Solar EnergySociety Annual Conf. Procs., Raleigh, N.C. (May 18, 2011), thedisclosures of which are incorporated by reference. As data are beingcollected, the researchers examine the data to determine if there areunderlying models that can translate results from these devices tophotovoltaic fleet production at a much broader area, yet fail toprovide translation of the data. In addition, half-hour or hourlysatellite irradiance data for specific locations and time periods ofinterest have been combined with randomly selected high speed data froma limited number of ground-based weather stations, such as described inCAISO 2011. “Summary of Preliminary Results of 33% Renewable IntegrationStudy—2010,” Cal. Public Util. Comm. LTPP No. R.10-05-006 (Apr. 29,2011) and J. Stein, “Simulation of 1-Minute Power Output fromUtility-Scale Photovoltaic Generation Systems,” Am. Solar Energy SocietyAnnual Conf. Procs., Raleigh, N.C. (May 18, 2011), the disclosures ofwhich are incorporated by reference. This approach, however, does notproduce time synchronized photovoltaic fleet variability for anyparticular time period because the locations of the ground-based weatherstations differ from the actual locations of the fleet. While suchresults may be useful as input data to photovoltaic simulation modelsfor purpose of performing high penetration photovoltaic studies, theyare not designed to produce data that could be used in grid operationaltools.

Similarly, accurate photovoltaic system configuration data is asimportant to photovoltaic power output forecasting as obtaining areliable solar irradiance forecast. The specification of a photovoltaicsystem's configuration are typically provided by the owner or operatorand can vary in terms of completeness, quality and correctness, whichcan complicate or skew power output forecasting. Moreover, in somesituations, photovoltaic system configuration specifications may simplynot be available, as can happen with privately-owned photovoltaicsystems. Residential systems, for example, are typically not controlledor accessible by the grid operators and power utility and otherpersonnel who need to fully understand and gauge their expectedphotovoltaic power output capabilities and shortcomings. Even largeutility-connected systems may have specifications that are not publiclyavailable due to privacy or security reasons.

Therefore, a need remains for an approach to determining photovoltaicsystem configuration specifications, even when configuration data isincomplete or unavailable, for use in forecasting power output.

SUMMARY

A computer-implemented system and method for inferring operationalspecifications of a photovoltaic power generation system using net loadis provided. Photovoltaic plant configuration specifications can beaccurately inferred with net load data and measured solar resource data.A time series of net load data is evaluated to identify, if possible, atime period with preferably minimum and consistent power consumption.Power generation data is simulated for a range of hypotheticalphotovoltaic system configurations based on a normalized solar powersimulation model. Net load data is estimated based on a base load and,if applicable, any binary loads and any variable loads. The set of keyparameters corresponding to the net load estimate that minimizes totalsquared error represents the inferred specifications of the photovoltaicplant configuration.

In one embodiment, a system and method for net load-based inference ofoperational specifications of a photovoltaic power generation systemwith the aid of a digital computer are provided. A time series of netload data for power consumed within a building measured over a timeperiod, the building also receiving power produced by a photovoltaicpower generation plant, is obtained. A time series of historicalmeasured irradiance data over the time period is obtained. A pluralityof photovoltaic plant configurations that each include a power ratingand operational features hypothesized for the plant are defined. Aplurality of key parameters are defined, the key parameters includingone or more component loads, the one or more component loads includingone of more of a base load of power consumed, at least one binary loadof power consumed within the building, and at least one variable load ofpower consumed within the building, wherein the base load represents aconstant power load drawn at all times within the building, the at leastone binary load represents a power load that is either on or off, andwhich, when on, draws power at a single power level within the building,and the at least one variable load represents a power load that can takeon multiple power levels within the building. Output production data foreach of the photovoltaic plant configurations is simulated based on anormalized photovoltaic power generation plant using the historicalmeasured irradiance data for the time period. Net load data associatedwith one or more of the photovoltaic plant configurations is estimatedby using the simulated power output production data for thatphotovoltaic plant configuration and the one or more of the componentloads. An error metric is calculated between the time series net loaddata and the estimated net load data associated with one or more of thephotovoltaic plant configurations. One of the photovoltaic plantconfigurations is inferred as a configuration of the plant based on theone or more error metrics. Some of the notable elements of thismethodology non-exclusively include:

(1) Employing a fully derived statistical approach to generatinghigh-speed photovoltaic fleet production data;

(2) Using a small sample of input data sources as diverse asground-based weather stations, existing photovoltaic systems, or solardata calculated from satellite images;

(3) Producing results that are usable for any photovoltaic fleetconfiguration;

(4) Supporting any time resolution, even those time resolutions fasterthan the input data collection rate;

(5) Providing results in a form that is useful and usable by electricpower grid planning and operation tools;

(6) Inferring photovoltaic plant configuration specifications, which canbe used to correct, replace or, if configuration data is unavailable,stand-in for the plant's specifications;

(7) Providing more accurate operational sets of photovoltaic systemspecifications to improve photovoltaic power generation fleetforecasting;

(8) Determining whether system maintenance is required or if degradationhas occurred by comparing reported power generation to expected powergeneration; and

(9) Quantifying the value of improving photovoltaic system performanceby modifying measured time series net load data using estimates of afully performing photovoltaic system and sending the results through autility bill analysis software program.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram showing a computer-implemented method forgenerating a probabilistic forecast of photovoltaic fleet powergeneration in accordance with one embodiment.

FIG. 2 is a block diagram showing a computer-implemented system forinferring operational specifications of a photovoltaic power generationsystem using net load in accordance with a further embodiment.

FIG. 3 is a graph depicting, by way of example, ten hours of time seriesirradiance data collected from a ground-based weather station with10-second resolution.

FIG. 4 is a graph depicting, by way of example, the clearness index thatcorresponds to the irradiance data presented in FIG. 3.

FIG. 5 is a graph depicting, by way of example, the change in clearnessindex that corresponds to the clearness index presented in FIG. 4.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5.

FIGS. 7A-7B are photographs showing, by way of example, the locations ofthe Cordelia Junction and Napa high density weather monitoring stations.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds.

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance.

FIGS. 10A-10F are graphs depicting, by way of example, the sameinformation as depicted in FIGS. 9A-9F versus temporal distance.

FIGS. 11A-11F are graphs depicting, by way of example, the predictedversus the measured variances of clearness indexes using differentreference time intervals.

FIGS. 12A-12F are graphs depicting, by way of example, the predictedversus the measured variances of change in clearness indexes usingdifferent reference time intervals.

FIGS. 13A-13F are graphs and a diagram depicting, by way of example,application of the methodology described herein to the Napa network.

FIG. 14 is a graph depicting, by way of example, an actual probabilitydistribution for a given distance between two pairs of locations, ascalculated for a 1,000 meter×1,000 meter grid in one square meterincrements.

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

FIG. 16 is a graph depicting, by way of example, results generated byapplication of Equation (65).

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions.

FIG. 18 is a graph depicting, by way of example, results by applicationof the model.

FIG. 19 is a flow diagram showing a computer-implemented method forinferring operational specifications of a photovoltaic power generationsystem in accordance with a further embodiment.

FIG. 20 is a flow diagram showing a routine 200 for simulating poweroutput of a photovoltaic power generation system 25 for use in themethod 180 of FIG. 19.

FIG. 21 is a table showing, by way of example, simulated half-hourphotovoltaic energy production for a 1-kW-AC photovoltaic system.

FIG. 22 are graphs depicting, by way of example, simulated versusmeasured power output for hypothetical photovoltaic system configurationspecifications evaluated using the method 180 of FIG. 19.

FIG. 23 is a graph depicting, by way of example, the relative meanabsolute error between the measured and simulated power output for allsystem configurations as shown in FIG. 22.

FIG. 24 are graphs depicting, by way of example, simulated versusmeasured power output for the optimal photovoltaic system configurationspecifications as shown in FIG. 22.

FIG. 25 is a flow diagram showing a computer-implemented method forinferring operational specifications of a photovoltaic power generationsystem using net load in accordance with a further embodiment.

FIG. 26 is a graph depicting, by way of example, energy consumption bythe exemplary house over a one-year period.

FIG. 27 is a graph depicting, by way of example, net load data for theexemplary house for a one-week period.

FIG. 28 is a graph depicting, by way of example, measured net load dataminus estimated base load data for the exemplary house for the one-weekperiod.

FIG. 29 is a graph depicting, by way of example, measured net load dataminus estimated base load data and estimated temperature-based attic fanload data for the exemplary house for the one-week period.

FIG. 30 is a graph depicting, by way of example, implied photovoltaicproduction compared to the simulated photovoltaic production for theexemplary house for the one-week period.

FIGS. 31A-F are graphs depicting, by way of example, comparing measuredand simulated net photovoltaic power production.

FIG. 32 is a graph depicting, by way of example, photovoltaic productionfor a four-year period.

DETAILED DESCRIPTION

Photovoltaic cells employ semiconductors exhibiting a photovoltaiceffect to generate direct current electricity through conversion ofsolar irradiance. Within each photovoltaic cell, light photons exciteelectrons in the semiconductors to create a higher state of energy,which acts as a charge carrier for the electrical current. The directcurrent electricity is converted by power inverters into alternatingcurrent electricity, which is then output for use in a power grid orother destination consumer. A photovoltaic system uses one or morephotovoltaic panels that are linked into an array to convert sunlightinto electricity. A single photovoltaic plant can include one or more ofthese photovoltaic arrays. In turn, a collection of photovoltaic plantscan be collectively operated as a photovoltaic fleet that is integratedinto a power grid, although the constituent photovoltaic plants withinthe fleet may actually be deployed at different physical locationsspread out over a geographic region.

To aid with the planning and operation of photovoltaic fleets, whetherat the power grid, supplemental, or standalone power generation levels,accurate photovoltaic system configuration specifications are needed toefficiently estimate individual photovoltaic power plant production.Photovoltaic system configurations can be inferred, even in the absenceof presumed configuration specifications, by evaluation of measuredhistorical photovoltaic system production data and measured historicalresource data. FIG. 1 is a flow diagram showing a computer-implementedmethod 10 for generating a probabilistic forecast of photovoltaic fleetpower generation in accordance with one embodiment. The method 10 can beimplemented in software and execution of the software can be performedon a computer system, such as further described infra, as a series ofprocess or method modules or steps.

A time series of solar irradiance or photovoltaic (“PV”) data is firstobtained (step 11) for a set of locations representative of thegeographic region within which the photovoltaic fleet is located orintended to operate, as further described infra with reference to FIG.3. Each time series contains solar irradiance observations measured orderived, then electronically recorded at a known sampling rate at fixedtime intervals, such as at half-hour intervals, over successiveobservational time periods. The solar irradiance observations caninclude solar irradiance measured by a representative set ofground-based weather stations (step 12), existing photovoltaic systems(step 13), satellite observations (step 14), or some combinationthereof. Other sources of the solar irradiance data are possible,including numeric weather prediction models.

Next, the solar irradiance data in the time series is converted overeach of the time periods, such as at half-hour intervals, into a set ofglobal horizontal irradiance clear sky indexes, which are calculatedrelative to clear sky global horizontal irradiance based on the type ofsolar irradiance data, such as described in commonly-assigned U.S.Patent application, entitled “Computer-Implemented Method for TuningPhotovoltaic Power Generation Plant Forecasting,” Ser. No. 13/677,175,filed Nov. 14, 2012, pending, the disclosure of which is incorporated byreference. The set of clearness indexes are interpreted into asirradiance statistics (step 15), as further described infra withreference to FIG. 4-6, and power statistics, including a time series ofthe power statistics for the photovoltaic plant, are generated (step 17)as a function of the irradiance statistics and photovoltaic plantconfiguration (step 16). The photovoltaic plant configuration includespower generation and location information, including direct current(“DC”) plant and photovoltaic panel ratings; number of power inverters;latitude, longitude and elevation; sampling and recording rates; sensortype, orientation, and number; voltage at point of delivery; trackingmode (fixed, single-axis tracking, dual-axis tracking), azimuth angle,tilt angle, row-to-row spacing, tracking rotation limit, and shading orother physical obstructions. Other types of information can also beincluded as part of the photovoltaic plant configuration. The resultanthigh-speed time series plant performance data can be combined toestimate photovoltaic fleet power output and variability, such asdescribed in commonly-assigned U.S. Pat. Nos. 8,165,811; 8,165,812;8,165,813; 8,326,535; 8,335,649; and 8,326,536, cited supra, for use bypower grid planners, operators and other interested parties.

The calculated irradiance statistics are combined with the photovoltaicfleet configuration to generate the high-speed time series photovoltaicproduction data. In a further embodiment, the foregoing methodology mayalso require conversion of weather data for a region, such as data fromsatellite regions, to average point weather data. A non-optimizedapproach would be to calculate a correlation coefficient matrixon-the-fly for each satellite data point. Alternatively, a conversionfactor for performing area-to-point conversion of satellite imagery datais described in commonly-assigned U.S. Pat. Nos. 8,165,813 and8,326,536, cited supra.

Each forecast of power production data for a photovoltaic plant predictsthe expected power output over a forecast period. FIG. 2 is a blockdiagram showing a computer-implemented system 20 for generating aprobabilistic forecast of photovoltaic fleet power generation inaccordance with one embodiment. Time series power output data 32 for aphotovoltaic plant is generated using observed field conditions relatingto overhead sky clearness. Solar irradiance 23 relative to prevailingcloudy conditions 22 in a geographic region of interest is measured.Direct solar irradiance measurements can be collected by ground-basedweather stations 24. Solar irradiance measurements can also be derivedor inferred by the actual power output of existing photovoltaic systems25. Additionally, satellite observations 26 can be obtained for thegeographic region. In a further embodiment, the solar irradiance can begenerated by numerical weather prediction models. Both the direct andinferred solar irradiance measurements are considered to be sets ofpoint values that relate to a specific physical location, whereassatellite imagery data is considered to be a set of area values thatneed to be converted into point values, such as described incommonly-assigned U.S. Pat. Nos. 8,165,813 and 8,326,536, cited supra.Still other sources of solar irradiance measurements are possible.

The solar irradiance measurements are centrally collected by a computersystem 21 or equivalent computational device. The computer system 21executes the methodology described supra with reference to FIG. 1 and asfurther detailed herein to generate time series power data 26 and otheranalytics, which can be stored or provided 27 to planners, operators,and other parties for use in solar power generation 28 planning andoperations. In a further embodiment, the computer system 21 executes themethodology described infra beginning with reference to FIG. 19 forinferring operational specifications of a photovoltaic power generationsystem, which can be stored or provided 27 to planners and otherinterested parties for use in predicting individual and fleet poweroutput generation. The data feeds 29 a-c from the various sources ofsolar irradiance data need not be high speed connections; rather, thesolar irradiance measurements can be obtained at an input datacollection rate and application of the methodology described hereinprovides the generation of an output time series at any time resolution,even faster than the input time resolution. The computer system 21includes hardware components conventionally found in a general purposeprogrammable computing device, such as a central processing unit,memory, user interfacing means, such as a keyboard, mouse, and display,input/output ports, network interface, and non-volatile storage, andexecute software programs structured into routines, functions, andmodules for execution on the various systems. In addition, otherconfigurations of computational resources, whether provided as adedicated system or arranged in client-server or peer-to-peertopologies, and including unitary or distributed processing,communications, storage, and user interfacing, are possible.

The detailed steps performed as part of the methodology described suprawith reference to FIG. 1 will now be described.

Obtain Time Series Irradiance Data

The first step is to obtain time series irradiance data fromrepresentative locations. This data can be obtained from ground-basedweather stations, existing photovoltaic systems, a satellite network, orsome combination sources, as well as from other sources. The solarirradiance data is collected from several sample locations across thegeographic region that encompasses the photovoltaic fleet.

Direct irradiance data can be obtained by collecting weather data fromground-based monitoring systems. FIG. 3 is a graph depicting, by way ofexample, ten hours of time series irradiance data collected from aground-based weather station with 10-second resolution, that is, thetime interval equals ten seconds. In the graph, the blue line 32 is themeasured horizontal irradiance and the black line 31 is the calculatedclear sky horizontal irradiance for the location of the weather station.

Irradiance data can also be inferred from select photovoltaic systemsusing their electrical power output measurements. A performance modelfor each photovoltaic system is first identified, and the input solarirradiance corresponding to the power output is determined.

Finally, satellite-based irradiance data can also be used. As satelliteimagery data is pixel-based, the data for the geographic region isprovided as a set of pixels, which span across the region andencompassing the photovoltaic fleet.

Calculate Irradiance Statistics

The time series irradiance data for each location is then converted intotime series clearness index data, which is then used to calculateirradiance statistics, as described infra.

Clearness Index (Kt)

The clearness index (Kt) is calculated for each observation in the dataset. In the case of an irradiance data set, the clearness index isdetermined by dividing the measured global horizontal irradiance by theclear sky global horizontal irradiance, may be obtained from any of avariety of analytical methods. FIG. 4 is a graph depicting, by way ofexample, the clearness index that corresponds to the irradiance datapresented in FIG. 3. Calculation of the clearness index as describedherein is also generally applicable to other expressions of irradianceand cloudy conditions, including global horizontal and direct normalirradiance.

Change in Clearness Index (ΔKt)

The change in clearness index (ΔKt) over a time increment of Δt is thedifference between the clearness index starting at the beginning of atime increment t and the clearness index starting at the beginning of atime increment t, plus a time increment Δt. FIG. 5 is a graph depicting,by way of example, the change in clearness index that corresponds to theclearness index presented in FIG. 4.

Time Period

The time series data set is next divided into time periods, forinstance, from five to sixty minutes, over which statisticalcalculations are performed. The determination of time period is selecteddepending upon the end use of the power output data and the timeresolution of the input data. For example, if fleet variabilitystatistics are to be used to schedule regulation reserves on a 30-minutebasis, the time period could be selected as 30 minutes. The time periodmust be long enough to contain a sufficient number of sampleobservations, as defined by the data time interval, yet be short enoughto be usable in the application of interest. An empirical investigationmay be required to determine the optimal time period as appropriate.

Fundamental Statistics

Table 1 lists the irradiance statistics calculated from time series datafor each time period at each location in the geographic region. Notethat time period and location subscripts are not included for eachstatistic for purposes of notational simplicity.

TABLE 1 Statistic Variable Mean clearness index μ_(Kt) Varianceclearness index σ_(Kt) ² Mean clearness index change μ_(ΔKt) Varianceclearness index change σ_(ΔKt) ²

Table 2 lists sample clearness index time series data and associatedirradiance statistics over five-minute time periods. The data is basedon time series clearness index data that has a one-minute time interval.The analysis was performed over a five-minute time period. Note that theclearness index at 12:06 is only used to calculate the clearness indexchange and not to calculate the irradiance statistics.

TABLE 2 Clearness Index Change Clearness Index (Kt) (ΔKt) 12:00 50% 40%12:01 90% 0% 12:02 90% −80% 12:03 10% 0% 12:04 10% 80% 12:05 90% −40%12:06 50% Mean (μ) 57% 0% Variance (σ²) 13% 27%

The mean clearness index change equals the first clearness index in thesucceeding time period, minus the first clearness index in the currenttime period divided by the number of time intervals in the time period.The mean clearness index change equals zero when these two values arethe same. The mean is small when there are a sufficient number of timeintervals. Furthermore, the mean is small relative to the clearnessindex change variance. To simplify the analysis, the mean clearnessindex change is assumed to equal zero for all time periods.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5 using a half-hour hour time period.Note that FIG. 6 presents the standard deviations, determined as thesquare root of the variance, rather than the variances, to present thestandard deviations in terms that are comparable to the mean.

Calculate Fleet Irradiance Statistics

Irradiance statistics were calculated in the previous section for thedata stream at each sample location in the geographic region. Themeaning of these statistics, however, depends upon the data source.Irradiance statistics calculated from a ground-based weather stationdata represent results for a specific geographical location as pointstatistics. Irradiance statistics calculated from satellite datarepresent results for a region as area statistics. For example, if asatellite pixel corresponds to a one square kilometer grid, then theresults represent the irradiance statistics across a physical area onekilometer square.

Average irradiance statistics across the photovoltaic fleet region are acritical part of the methodology described herein. This section presentsthe steps to combine the statistical results for individual locationsand calculate average irradiance statistics for the region as a whole.The steps differ depending upon whether point statistics or areastatistics are used.

Irradiance statistics derived from ground-based sources simply need tobe averaged to form the average irradiance statistics across thephotovoltaic fleet region. Irradiance statistics from satellite sourcesare first converted from irradiance statistics for an area intoirradiance statistics for an average point within the pixel. The averagepoint statistics are then averaged across all satellite pixels todetermine the average across the photovoltaic fleet region.

Mean Clearness Index (μ _(Kt) ) and Mean Change in Clearness Index (μ_(ΔKt) )

The mean clearness index should be averaged no matter what input datasource is used, whether ground, satellite, or photovoltaic systemoriginated data. If there are N locations, then the average clearnessindex across the photovoltaic fleet region is calculated as follows.

$\begin{matrix}{\mu_{\overset{\_}{Kt}} = {\sum\limits_{i = 1}^{N}\; \frac{\mu_{{Kt}_{i}}}{N}}} & (1)\end{matrix}$

The mean change in clearness index for any period is assumed to be zero.As a result, the mean change in clearness index for the region is alsozero.

α _(ΔKt)   (2)

Convert Area Variance to Point Variance

The following calculations are required if satellite data is used as thesource of irradiance data. Satellite observations represent valuesaveraged across the area of the pixel, rather than single pointobservations. The clearness index derived from this data (Kt^(Area))(Kt^(Area)) may therefore be considered an average of many individualpoint measurements.

$\begin{matrix}{{Kt}^{Area} = {\sum\limits_{i = 1}^{N}\frac{{Kt}^{i}}{N}}} & (3)\end{matrix}$

As a result, the variance of the area clearness index based on satellitedata can be expressed as the variance of the average clearness indexesacross all locations within the satellite pixel.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{{VAR}\left\lbrack {Kt}^{Area} \right\rbrack} = {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}\; \frac{{Kt}^{i}}{N}} \right\rbrack}}} & (4)\end{matrix}$

The variance of a sum, however, equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\; {{COV}\left\lbrack {{Kt}^{i},{Kt}^{j}} \right\rbrack}}}}} & (9)\end{matrix}$

Let ρ^(Kt) ^(i) ^(,Kt) ^(j) represents the correlation coefficientbetween the clearness index at location i and location j within thesatellite pixel. By definition of correlation coefficient,COV[Kt^(i),Kt^(j)]=σ_(Kt) ^(i)σ_(Kt) ^(j)σ^(Kt) ^(i) ^(,Kt) ^(j) .Furthermore, since the objective is to determine the average pointvariance across the satellite pixel, the standard deviation at any pointwithin the satellite pixel can be assumed to be the same and equalsσ_(Kt), which means that σ_(Kt) ^(i)σ_(Kt) ^(j)=_(Kt) ² for all locationpairs. As a result, COV[Kt^(i),Kt^(j)]=σ_(Kt) ²ρ^(Kt) ^(i) ^(,Kt) ^(j) .Substituting this result into Equation (5) and simplify.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{\sigma_{Kt}^{2}\left( \frac{1}{N^{2}} \right)}{\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\rho^{{Kt}^{i},{Kt}^{j}}}}}} & (6)\end{matrix}$

Suppose that data was available to calculate the correlation coefficientin Equation (6). The computational effort required to perform a doublesummation for many points can be quite large and computationallyresource intensive. For example, a satellite pixel representing a onesquare kilometer area contains one million square meter increments. Withone million increments, Equation (6) would require one trillioncalculations to compute.

The calculation can be simplified by conversion into a continuousprobability density function of distances between location pairs acrossthe pixel and the correlation coefficient for that given distance, asfurther described supra. Thus, the irradiance statistics for a specificsatellite pixel, that is, an area statistic, rather than a pointstatistics, can be converted into the irradiance statistics at anaverage point within that pixel by dividing by a “Area” term (A), whichcorresponds to the area of the satellite pixel. Furthermore, theprobability density function and correlation coefficient functions aregenerally assumed to be the same for all pixels within the fleet region,making the value of A constant for all pixels and reducing thecomputational burden further. Details as to how to calculate A are alsofurther described supra.

$\begin{matrix}{\sigma_{Kt}^{2} = \frac{\sigma_{{Kt} - {Area}}^{2}}{A_{Kt}^{{Satellite}\mspace{14mu} {Pixel}}}} & (7)\end{matrix}$

where:

$\begin{matrix}{A_{Kt}^{{Satellite}\mspace{14mu} {Pixel}} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (8)\end{matrix}$

Likewise, the change in clearness index variance across the satelliteregion can also be converted to an average point estimate using asimilar conversion factor, A_(ΔKt) ^(Area).

$\begin{matrix}{\sigma_{\Delta \; {Kt}}^{2} = \frac{\sigma_{{\Delta \; {Kt}} - {Area}}^{2}}{A_{\Delta \; {Kt}}^{{Satellite}\mspace{14mu} {Pixel}}}} & (9)\end{matrix}$

Variance of Clearness Index (σ _(Kt) ²) and Variance of Change inClearness Index (σ _(ΔKt) ²)

At this point, the point statistics (σ_(Kt) ² and σ_(ΔKt) ²) have beendetermined for each of several representative locations within the fleetregion. These values may have been obtained from either ground-basedpoint data or by converting satellite data from area into pointstatistics. If the fleet region is small, the variances calculated ateach location i can be averaged to determine the average point varianceacross the fleet region. If there are N locations, then average varianceof the clearness index across the photovoltaic fleet region iscalculated as follows.

$\begin{matrix}{\sigma^{\frac{2}{Kt}} = {\sum\limits_{i = 1}^{N}\; \frac{\sigma_{{Kt}_{i}}^{2}}{N}}} & (10)\end{matrix}$

Likewise, the variance of the clearness index change is calculated asfollows.

$\begin{matrix}{\sigma^{\frac{2}{\Delta \; {Kt}}} = {\sum\limits_{i = 1}^{N}\; \frac{\sigma_{\Delta \; {Kt}_{i}}^{2}}{N}}} & (11)\end{matrix}$

Calculate Fleet Power Statistics

The next step is to calculate photovoltaic fleet power statistics usingthe fleet irradiance statistics, as determined supra, and physicalphotovoltaic fleet configuration data. These fleet power statistics arederived from the irradiance statistics and have the same time period.

The critical photovoltaic fleet performance statistics that are ofinterest are the mean fleet power, the variance of the fleet power, andthe variance of the change in fleet power over the desired time period.As in the case of irradiance statistics, the mean change in fleet poweris assumed to be zero.

Photovoltaic System Power for Single System at Time t Photovoltaicsystem power output (kW) is approximately linearly related to theAC-rating of the photovoltaic system (R in units of kW_(AC)) timesplane-of-array irradiance. Plane-of-array irradiance can be representedby the clearness index over the photovoltaic system (KtPV) times theclear sky global horizontal irradiance times an orientation factor (O),which both converts global horizontal irradiance to plane-of-arrayirradiance and has an embedded factor that converts irradiance fromWatts/m² to kW output/kW of rating. Thus, at a specific point in time(t), the power output for a single photovoltaic system (n) equals:

P _(t) ^(n) =R ^(n) O _(t) ^(n) KtPV_(t) ^(n) I _(t) ^(Clear,n)  (12)

The change in power equals the difference in power at two differentpoints in time.

ΔP _(t,Δt) ^(n) =R ^(n) O _(t+Δt) ^(n) KtPV_(t+Δt) ^(n) I _(t+Δt)^(Clear,n) −R ^(n) O _(t) ^(n) KtPV_(t) ^(n) I _(t) ^(Clear,n)  (13)

The rating is constant, and over a short time interval, the two clearsky plane-of-array irradiances are approximately the same (O_(t+Δt)^(n)I_(t+Δt) ^(Clear,n)≈O_(t) ^(n)I_(t) ^(Clear,n)) so that the threeterms can be factored out and the change in the clearness index remains.

ΔP _(t,Δt) ^(n) ≈R ^(n) O _(t) ^(n) I _(t) ^(Clear,n) ΔKtPV_(t)^(n)  (14)

Time Series Photovoltaic Power for Single System

P^(n) is a random variable that summarizes the power for a singlephotovoltaic system n over a set of times for a given time interval andset of time periods. ΔP^(n) is a random variable that summarizes thechange in power over the same set of times.

Mean Fleet Power (μ_(P))

The mean power for the fleet of photovoltaic systems over the timeperiod equals the expected value of the sum of the power output from allof the photovoltaic systems in the fleet.

$\begin{matrix}{\mu_{P} = {E\left\lbrack {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (15)\end{matrix}$

If the time period is short and the region small, the clear skyirradiance does not change much and can be factored out of theexpectation.

$\begin{matrix}{\mu_{P} = {\mu_{I^{Clear}}{E\left\lbrack {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}{KtPV}^{n}}} \right\rbrack}}} & (16)\end{matrix}$

Again, if the time period is short and the region small, the clearnessindex can be averaged across the photovoltaic fleet region and any givenorientation factor can be assumed to be a constant within the timeperiod. The result is that:

μ_(P) =R ^(Adj.Fleet)μ_(I) _(Clear) μ _(Kt)   (17)

where μ_(I) _(Clear) is calculated, μ _(Kt) is taken from Equation (1)and:

$\begin{matrix}{R^{{Adj}.{Fleet}} = {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}}}} & (18)\end{matrix}$

This value can also be expressed as the average power during clear skyconditions times the average clearness index across the region.

μ_(P)=μ_(P) _(Clear) μ _(Kt)   (19)

Variance of Fleet Power (σ_(P) ²)

The variance of the power from the photovoltaic fleet equals:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (20)\end{matrix}$

If the clear sky irradiance is the same for all systems, which will bethe case when the region is small and the time period is short, then:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {I^{Clear}{\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}{KtPV}^{n}}}} \right\rbrack}} & (21)\end{matrix}$

The variance of a product of two independent random variables X, Y, thatis, VAR[XY]) equals E[X]²VAR[Y]+E[Y]²VAR[X]+VAR[X]VAR[Y]. If the Xrandom variable has a large mean and small variance relative to theother terms, then VAR[XV]≈E[X]²VAR[Y]. Thus, the clear sky irradiancecan be factored out of Equation (21) and can be written as:

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}{{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}\; {R^{n}{KtPV}^{n}O^{n}}} \right\rbrack}}} & (22)\end{matrix}$

The variance of a sum equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\; {{COV}\left\lbrack {{R^{i}{KtPV}^{i}O^{i}},{R^{j}{KtPV}^{j}O^{j}}} \right\rbrack}}} \right)}} & (23)\end{matrix}$

In addition, over a short time period, the factor to convert from clearsky GHI to clear sky POA does not vary much and becomes a constant. Allfour variables can be factored out of the covariance equation.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\; {\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right){{COV}\left\lbrack {{KtPV}^{i},{KtPV}^{j}} \right\rbrack}}}} \right)}} & (24)\end{matrix}$

For any i and j, COV[KtPV^(i),KtPV^(j)]=√{square root over (σ_(KtPV)_(i) ²σ_(KtPV) _(j) ²)}ρ^(Kt) ^(i) ^(,Kt) ^(j) .

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right)\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}\rho^{{Kt}^{i},{Kt}^{j}}}}}} \right)}} & (25)\end{matrix}$

As discussed supra, the variance of the satellite data required aconversion from the satellite area, that is, the area covered by apixel, to an average point within the satellite area. In the same way,assuming a uniform clearness index across the region of the photovoltaicplant, the variance of the clearness index across a region the size ofthe photovoltaic plant within the fleet also needs to be adjusted. Thesame approach that was used to adjust the satellite clearness index canbe used to adjust the photovoltaic clearness index. Thus, each varianceneeds to be adjusted to reflect the area that the i^(th) photovoltaicplant covers.

$\begin{matrix}{\sigma_{{KtPV}^{i}}^{2} = {A_{Kt}^{i}\sigma \frac{2}{Kt}}} & (26)\end{matrix}$

Substituting and then factoring the clearness index variance given theassumption that the average variance is constant across the regionyields:

$\begin{matrix}{\sigma_{P}^{2} = {\left( {R^{{Adj}.{Fleet}}\mu_{I^{Clear}}} \right)^{2}P^{Kt}\sigma \frac{2}{Kt}}} & (27)\end{matrix}$

where the correlation matrix equals:

$\begin{matrix}{P^{Kt} = \frac{\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\; {\left( {R^{i}O^{i}A_{Kt}^{i}} \right)\left( {R^{j}O^{j}A_{Kt}^{j}} \right)\rho^{{Kt}^{i},{Kt}^{j}}}}}{\left( {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}}} \right)^{2}}} & (28)\end{matrix}$

R^(Adj.Fleet)μ_(I) _(Clear) in Equation (27) can be written as the powerproduced by the photovoltaic fleet under clear sky conditions, that is:

$\begin{matrix}{\sigma_{P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{Kt}\sigma \frac{2}{\Delta \; {Kt}}}} & (29)\end{matrix}$

If the region is large and the clearness index mean or variances varysubstantially across the region, then the simplifications may not beable to be applied. Notwithstanding, if the simplification isinapplicable, the systems are likely located far enough away from eachother, so as to be independent. In that case, the correlationcoefficients between plants in different regions would be zero, so mostof the terms in the summation are also zero and an inter-regionalsimplification can be made. The variance and mean then become theweighted average values based on regional photovoltaic capacity andorientation.

DISCUSSION

In Equation (28), the correlation matrix term embeds the effect ofintra-plant and inter-plant geographic diversification. The area-relatedterms (A) inside the summations reflect the intra-plant power smoothingthat takes place in a large plant and may be calculated using thesimplified relationship, as further discussed supra. These terms arethen weighted by the effective plant output at the time, that is, therating adjusted for orientation. The multiplication of these terms withthe correlation coefficients reflects the inter-plant smoothing due tothe separation of photovoltaic systems from one another.

Variance of Change in Fleet Power (σ_(ΔP) ²)

A similar approach can be used to show that the variance of the changein power equals:

$\begin{matrix}{\sigma_{\Delta \; P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{\Delta \; {Kt}}\sigma \frac{2}{\Delta \; {Kt}}}} & (30)\end{matrix}$

where:

$\begin{matrix}{P^{\Delta \; {Kt}} = \frac{\sum\limits_{i = 1}^{N}\; {\sum\limits_{j = 1}^{N}\; {\left( {R^{i}O^{i}A_{\Delta \; {Kt}}^{i}} \right)\left( {R^{j}O^{j}A_{\Delta \; {Kt}}^{j}} \right)\rho^{{\Delta \; {Kt}^{i}},{\Delta \; {Kt}^{j}}}}}}{\left( {\sum\limits_{n = 1}^{N}\; {R^{n}O^{n}}} \right)^{2}}} & (31)\end{matrix}$

The determination of Equations (30) and (31) becomes computationallyintensive as the network of points becomes large. For example, a networkwith 10,000 photovoltaic systems would require the computation of acorrelation coefficient matrix with 100 million calculations. Thecomputational burden can be reduced in two ways. First, many of theterms in the matrix are zero because the photovoltaic systems arelocated too far away from each other. Thus, the double summation portionof the calculation can be simplified to eliminate zero values based ondistance between locations by construction of a grid of points. Second,once the simplification has been made, rather than calculating thematrix on-the-fly for every time period, the matrix can be calculatedonce at the beginning of the analysis for a variety of cloud speedconditions, and then the analysis would simply require a lookup of theappropriate value.

Time Lag Correlation Coefficient

The next step is to adjust the photovoltaic fleet power statistics fromthe input time interval to the desired output time interval. Forexample, the time series data may have been collected and stored every60 seconds. The user of the results, however, may want to havephotovoltaic fleet power statistics at a 10-second rate. This adjustmentis made using the time lag correlation coefficient.

The time lag correlation coefficient reflects the relationship betweenfleet power and that same fleet power starting one time interval (Δt)later. Specifically, the time lag correlation coefficient is defined asfollows:

$\begin{matrix}{\rho^{P,P^{\Delta \; t}} = \frac{{COV}\left\lbrack {P,P^{\Delta \; t}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta \; t}}^{2}}}} & (32)\end{matrix}$

The assumption that the mean clearness index change equals zero impliesthat σ_(P) _(Δt) ²=σ_(P) ². Given a non-zero variance of power, thisassumption can also be used to show that Therefore:

$\frac{{COV}\left\lbrack {P,P^{\Delta \; t}} \right\rbrack}{\sigma_{P}^{2}} = {1 - {\frac{\sigma_{\Delta \; P}^{2}}{2\; \sigma_{P}^{2}}.}}$

$\begin{matrix}{\rho^{P,P^{\Delta \; t}} = {1 - \frac{\sigma_{\Delta \; P}^{2}}{2\; \sigma_{P}^{2}}}} & (33)\end{matrix}$

This relationship illustrates how the time lag correlation coefficientfor the time interval associated with the data collection rate iscompletely defined in terms of fleet power statistics alreadycalculated. A more detailed derivation is described infra.

Equation (33) can be stated completely in terms of the photovoltaicfleet configuration and the fleet region clearness index statistics bysubstituting Equations (29) and (30). Specifically, the time lagcorrelation coefficient can be stated entirely in terms of photovoltaicfleet configuration, the variance of the clearness index, and thevariance of the change in the clearness index associated with the timeincrement of the input data.

$\begin{matrix}{\rho^{P,P^{\Delta \; t}} = {1 - \frac{P^{\Delta \; K\; t}\sigma \frac{2}{\Delta \; {Kt}}}{2\; P^{K\; t}\sigma \frac{2}{Kt}}}} & (34)\end{matrix}$

Generate High-Speed Time Series Photovoltaic Fleet Power

The final step is to generate high-speed time series photovoltaic fleetpower data based on irradiance statistics, photovoltaic fleetconfiguration, and the time lag correlation coefficient. This step is toconstruct time series photovoltaic fleet production from statisticalmeasures over the desired time period, for instance, at half-hour outputintervals.

A joint probability distribution function is required for this step. Thebivariate probability density function of two unit normal randomvariables (X and Y) with a correlation coefficient of ρ equals:

$\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{2\; \pi \sqrt{1 - \rho^{2}}}{\exp \left\lbrack {- \frac{\left( {x^{2} + y^{2} - {2\; \rho \; {xy}}} \right)}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (35)\end{matrix}$

The single variable probability density function for a unit normalrandom variable X alone is

$(x) = {\frac{1}{\sqrt{2\; \pi}}{{\exp \left( {- \frac{x^{2}}{2}} \right)}.}}$

In addition, a conditional distribution for y can be calculated based ona known x by dividing the bivariate probability density function by thesingle variable probability density, that is,

${f\left( {yx} \right)} = {\frac{f\left( {x,y} \right)}{f(x)}.}$

Making the appropriate substitutions, the result is that the conditionaldistribution of y based on a known x equals:

$\begin{matrix}{{f\left( {yx} \right)} = {\frac{1}{\sqrt{2\; \pi}\sqrt{1 - \rho^{2}}}{\exp \left\lbrack {- \frac{\left( {y - {\rho \; x}} \right)^{2}}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (36)\end{matrix}$

Define a random variable

$Z = \frac{Y - {\rho \; x}}{\sqrt{1 - \rho^{2}}}$

and substitute into Equation (36).

The result is that the conditional probability of z given a known xequals:

$\begin{matrix}{{f\left( z \middle| x \right)} = {\frac{1}{\sqrt{2\pi}}{\exp\left( {- \frac{z^{2}}{2}} \right)}}} & (37)\end{matrix}$

The cumulative distribution function for Z can be denoted by Φ(z*),where z* represents a specific value for z. The result equals aprobability (p) that ranges between 0 (when z*=−∞) and 1 (when z*=∞).The function represents the cumulative probability that any value of zis less than z*, as determined by a computer program or value lookup.

$\begin{matrix}{p = {{\Phi \left( z^{*} \right)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{z^{*}}{{\exp\left( {- \frac{z^{2}}{2}} \right)}\ {dz}}}}}} & (38)\end{matrix}$

Rather than selecting z*, however, a probability p falling between 0 and1 can be selected and the corresponding z* that results in thisprobability found, which can be accomplished by taking the inverse ofthe cumulative distribution function.

Φ⁻¹(p)=z*  (39)

Substituting back for z as defined above results in:

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{y - {\rho \; x}}{\sqrt{1 - \rho^{2}}}} & (40)\end{matrix}$

Now, let the random variables equal

${X = {{\frac{P - \mu_{P}}{\sigma_{P}}\mspace{14mu} {and}\mspace{14mu} Y} = \frac{P^{\Delta \; t} - \mu_{P^{\Delta \; t}}}{\sigma_{P^{\Delta \; t}}}}},$

with the correlation coefficient being the time lag correlationcoefficient between P and P^(Δt), that is, let ρ=ρ^(P,P) ^(Δt) . When Δtis small, then the mean and standard deviations for P^(Δt) areapproximately equal to the mean and standard deviation for P. Thus, Ycan be restated as

$Y \approx {\frac{P^{\Delta \; t} - \mu_{P}}{\sigma_{P}}.}$

Add a time subscript to all of the relevant data to represent a specificpoint in time and substitute x, y, and ρ into Equation (40).

$\begin{matrix}{{\Phi^{- 1}(p)} = \frac{\left( \frac{P_{t}^{\Delta \; t} - \mu_{P}}{\sigma_{P}} \right) - {\rho^{P,P^{\Delta \; t}}\left( \frac{P_{t} - \mu_{P}}{\sigma_{P}} \right)}}{\sqrt{1 - \rho^{P,P^{\Delta \; t^{2}}}}}} & (41)\end{matrix}$

The random variable P^(Δ1), however, is simply the random variable Pshifted in time by a time interval of Δt. As a result, at any given timet, P^(Δt) _(t)=P_(t+Δt). Make this substitution into Equation (41) andsolve in terms of P_(t+Δt).

$\begin{matrix}{P_{t + {\Delta \; t}} = {{\rho^{P,P^{\Delta \; t}}P_{t}} + {\left( {1 - \rho^{P,P^{\Delta \; t}}} \right)\mu_{P}} + {\sqrt{\sigma_{P}^{2}\left( {1 - \rho^{P,P^{\Delta \; t^{2}}}} \right)}{\Phi^{- 1}(p)}}}} & (42)\end{matrix}$

At any given time, photovoltaic fleet power equals photovoltaic fleetpower under clear sky conditions times the average regional clearnessindex, that is, P_(t)=P_(t) ^(Clear)Kt_(t). In addition, over a shorttime period, μ_(P)≈P_(t) ^(Clear)μ _(Kt) and

$\sigma_{P}^{2} \approx {\left( P_{t}^{Clear} \right)^{2}\mspace{11mu} P^{K\; t}{\sigma_{\overset{\_}{Kt}}^{2}.}}$

Substitute these three relationships into Equation (42) and factor outphotovoltaic fleet power under clear sky conditions (P_(t) ^(Clear)) ascommon to all three terms.

$\begin{matrix}{P_{t + {\Delta \; t}} = {P_{t}^{Clear}{\quad\left\lbrack {{\rho^{P,P^{\Delta \; t}}{Kt}_{t}} + {\left( {1 - \rho^{P,P^{\Delta \; t}}} \right)\mu_{\overset{\_}{Kt}}} + {\sqrt{P^{Kt}\sigma \frac{2}{Kt}\left( {1 - \rho^{P,P^{\Delta \; t^{2}}}} \right)}{\Phi^{- 1}\left( p_{t} \right)}}} \right\rbrack}}} & (43)\end{matrix}$

Equation (43) provides an iterative method to generate high-speed timeseries photovoltaic production data for a fleet of photovoltaic systems.At each time step (t+Δt), the power delivered by the fleet ofphotovoltaic systems (P_(t+Δt)) is calculated using input values fromtime step t. Thus, a time series of power outputs can be created. Theinputs include:

-   -   P_(t) ^(Clear)—photovoltaic fleet power during clear sky        conditions calculated using a photovoltaic simulation program        and clear sky irradiance.    -   Kt_(t)—average regional clearness index inferred based on P_(t)        calculated in time step t, that is, Kt_(t)=P_(t)/P_(t) ^(Clear).    -   μ _(Kt) —mean clearness index calculated using time series        irradiance data and Equation (1).    -   σ _(Kt) ²—variance of the clearness index calculated using time        series irradiance data and Equation (10).    -   ρ^(P,P) ^(Δt) —fleet configuration as reflected in the time lag        correlation coefficient calculated using Equation (34). In turn,        Equation (34), relies upon correlation coefficients from        Equations (28) and (31). A method to obtain these correlation        coefficients by empirical means is described in        commonly-assigned U.S. Pat. No. 8,165,811, issued Apr. 24, 2012,        and U.S. Pat. No. 8,165,813, issued Apr. 24, 2012, the        disclosure of which are incorporated by reference.    -   P^(Kt)—fleet configuration as reflected in the clearness index        correlation coefficient matrix calculated using Equation (28)        where, again, the correlation coefficients may be obtained using        the empirical results as further described infra.    -   Φ⁻¹(p_(t))—the inverse cumulative normal distribution function        based on a random variable between 0 and 1.

Derivation of Empirical Models

The previous section developed the mathematical relationships used tocalculate irradiance and power statistics for the region associated witha photovoltaic fleet. The relationships between Equations (8), (28),(31), and (34) depend upon the ability to obtain point-to-pointcorrelation coefficients. This section presents empirically-derivedmodels that can be used to determine the value of the coefficients forthis purpose.

A mobile network of 25 weather monitoring devices was deployed in a 400meter by 400 meter grid in Cordelia Junction, Calif., between Nov. 6,2010, and Nov. 15, 2010, and in a 4,000 meter by 4,000 meter grid inNapa, Calif., between Nov. 19, 2010, and Nov. 24, 2010. FIGS. 7A-7B arephotographs showing, by way of example, the locations of the CordeliaJunction and Napa high density weather monitoring stations.

An analysis was performed by examining results from Napa and CordeliaJunction using 10, 30, 60, 120 and 180 second time intervals over eachhalf-hour time period in the data set. The variance of the clearnessindex and the variance of the change in clearness index were calculatedfor each of the 25 locations for each of the two networks. In addition,the clearness index correlation coefficient and the change in clearnessindex correlation coefficient for each of the 625 possible pairs, 300 ofwhich are unique, for each of the two locations were calculated.

An empirical model is proposed as part of the methodology describedherein to estimate the correlation coefficient of the clearness indexand change in clearness index between any two points by using as inputsthe following: distance between the two points, cloud speed, and timeinterval. For the analysis, distances were measured, cloud speed wasimplied, and a time interval was selected.

The empirical models infra describe correlation coefficients between twopoints (i and j), making use of “temporal distance,” defined as thephysical distance (meters) between points i and j, divided by theregional cloud speed (meters per second) and having units of seconds.The temporal distance answers the question, “How much time is needed tospan two locations?”

Cloud speed was estimated to be six meters per second. Results indicatethat the clearness index correlation coefficient between the twolocations closely matches the estimated value as calculated using thefollowing empirical model:

ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C₁×TemporalDistance)^(ClearnessPower)  (44)

where TemporalDistance=Distance (meters)/CloudSpeed (meters per second),ClearnessPower=ln(C₂Δt)−9.3, such that 5≤k≤15, where the expected valueis k=9.3, Δt is the desired output time interval (seconds), C₁=10⁻³seconds⁻¹, and C₂=1 seconds⁻¹.

Results also indicate that the correlation coefficient for the change inclearness index between two locations closely matches the valuescalculated using the following empirical relationship:

ρ^(ΔKt) ^(i) ^(,ΔKt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j))^(ΔClearnessPower)  (45)

where ρ^(Kt) ^(i) ^(,Kt) ^(j) is calculated using Equation (44) and

${{\Delta \; {ClearnessPower}} = {1 + \frac{140}{C_{2}\Delta \; t}}},$

such that 100≤m≤200, where the expected value is m=140.

Empirical results also lead to the following models that may be used totranslate the variance of clearness index and the variance of change inclearness index from the measured time interval (Δt ref) to the desiredoutput time interval (Δt).

$\begin{matrix}{\sigma_{{Kt}_{\Delta \; t}}^{2} = {\sigma_{{Kt}_{\Delta \; t\mspace{11mu} {ref}}}^{2}{\exp \left\lbrack {1 - \left( \frac{\Delta \; t}{\Delta \; t\mspace{11mu} {ref}} \right)^{C_{3}}} \right\rbrack}}} & (46) \\{\sigma_{\Delta \; {Kt}_{\Delta \; t}}^{2} = {\sigma_{\Delta \; {Kt}_{\Delta \; t\mspace{11mu} {ref}}}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta \; t}{\Delta \; t\mspace{11mu} {ref}} \right)^{C_{3}}} \right\rbrack}} \right\}}} & (47)\end{matrix}$

where C₃=0.1≤C₃≤0.2, where the expected value is C₃=0.15.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds. Forexample, if the variance is calculated at a 300-second time interval andthe user desires results at a 10-second time interval, the adjustmentfor the variance clearness index would be 1.49

These empirical models represent a valuable means to rapidly calculatecorrelation coefficients and translate time interval withreadily-available information, which avoids the use ofcomputation-intensive calculations and high-speed streams of data frommany point sources, as would otherwise be required.

Validation

Equations (44) and (45) were validated by calculating the correlationcoefficients for every pair of locations in the Cordelia Junctionnetwork and the Napa network at half-hour time periods. The correlationcoefficients for each time period were then weighted by thecorresponding variance of that location and time period to determineweighted average correlation coefficient for each location pair. Theweighting was performed as follows:

${\overset{\_}{{\rho^{{Kt}^{\iota},{Kt}}}^{J}} = \frac{\sum\limits_{t = 1}^{T}{\sigma_{{{Kt} - i},j_{t}}^{2}\rho^{{Kt}^{i},{Kt}^{j_{t}}}}}{\sum\limits_{t = 1}^{T}\sigma_{{{Kt} - i},j_{t}}^{2}}},{and}$$\overset{\_}{{\rho^{{\Delta \; {Kt}^{\iota}},{\Delta \; {Kt}}}}^{J}} = {\frac{{\sum\limits_{t = 1}^{T}\sigma_{{\Delta \; {Kt}} - i}^{2}},{j_{t}\rho^{{\Delta \; {Kt}^{i}},{\Delta \; {Kt}^{j_{t}}}}}}{\sum\limits_{t = 1}^{T}\sigma_{{{\Delta \; {Kt}} - i},j_{t}}^{2}}.}$

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance. FIGS. 10A-10F are graphs depicting, by way ofexample, the same information as depicted in FIGS. 9A-9F versus temporaldistance, based on the assumption that cloud speed was 6 meters persecond. The upper line and dots appearing in close proximity to theupper line present the clearness index and the lower line and dotsappearing in close proximity to the lower line present the change inclearness index for time intervals from 10 seconds to 5 minutes. Thesymbols are the measured results and the lines are the predictedresults.

Several observations can be drawn based on the information provided bythe FIGS. 9A-9F and 10A-10F. First, for a given time interval, thecorrelation coefficients for both the clearness index and the change inthe clearness index follow an exponential decline pattern versusdistance (and temporal distance). Second, the predicted results are agood representation of the measured results for both the correlationcoefficients and the variances, even though the results are for twoseparate networks that vary in size by a factor of 100. Third, thechange in the clearness index correlation coefficient converges to theclearness correlation coefficient as the time interval increases. Thisconvergence is predicted based on the form of the empirical modelbecause ΔClearnessPower approaches one as Δt becomes large.

Equations (46) and (47) were validated by calculating the averagevariance of the clearness index and the variance of the change in theclearness index across the 25 locations in each network for everyhalf-hour time period. FIGS. 11A-11F are graphs depicting, by way ofexample, the predicted versus the measured variances of clearnessindexes using different reference time intervals. FIGS. 12A-12F aregraphs depicting, by way of example, the predicted versus the measuredvariances of change in clearness indexes using different reference timeintervals. FIGS. 11A-11F and 12A-12F suggest that the predicted resultsare similar to the measured results.

DISCUSSION

The point-to-point correlation coefficients calculated using theempirical forms described supra refer to the locations of specificphotovoltaic power production sites. Importantly, note that the dataused to calculate these coefficients was not obtained from time sequencemeasurements taken at the points themselves. Rather, the coefficientswere calculated from fleet-level data (cloud speed), fixed fleet data(distances between points), and user-specified data (time interval).

The empirical relationships of the foregoing types of empiricalrelationships may be used to rapidly compute the coefficients that arethen used in the fundamental mathematical relationships. The methodologydoes not require that these specific empirical models be used andimproved models will become available in the future with additional dataand analysis.

Example

This section provides a complete illustration of how to apply themethodology using data from the Napa network of 25 irradiance sensors onNov. 21, 2010. In this example, the sensors served as proxies for anactual 1-kW photovoltaic fleet spread evenly over the geographicalregion as defined by the sensors. For comparison purposes, a directmeasurement approach is used to determine the power of this fleet andthe change in power, which is accomplished by adding up the 10-secondoutput from each of the sensors and normalizing the output to a 1-kWsystem. FIGS. 13A-13F are graphs and a diagram depicting, by way ofexample, application of the methodology described herein to the Napanetwork.

The predicted behavior of the hypothetical photovoltaic fleet wasseparately estimated using the steps of the methodology described supra.The irradiance data was measured using ground-based sensors, althoughother sources of data could be used, including from existingphotovoltaic systems or satellite imagery. As shown in FIG. 13A, thedata was collected on a day with highly variable clouds with one-minuteglobal horizontal irradiance data collected at one of the 25 locationsfor the Napa network and specific 10-second measured power outputrepresented by a blue line. This irradiance data was then converted fromglobal horizontal irradiance to a clearness index. The mean clearnessindex, variance of clearness index, and variance of the change inclearness index was then calculated for every 15-minute period in theday. These calculations were performed for each of the 25 locations inthe network. Satellite-based data or a statistically-significant subsetof the ground measurement locations could have also served in place ofthe ground-based irradiance data. However, if the data had beencollected from satellite regions, an additional translation from areastatistics to average point statistics would have been required. Theaveraged irradiance statistics from Equations (1), (10), and (11) areshown in FIG. 13B, where standard deviation (σ) is presented, instead ofvariance (σ²) to plot each of these values in the same units.

In this example, the irradiance statistics need to be translated sincethe data were recorded at a time interval of 60 seconds, but the desiredresults are at a 10-second resolution. The translation was performedusing Equations (46) and (47) and the result is presented in FIG. 13C.

The details of the photovoltaic fleet configuration were then obtained.The layout of the fleet is presented in FIG. 13D. The details includethe location of the each photovoltaic system (latitude and longitude),photovoltaic system rating (1/25 kW), and system orientation (all arehorizontal).

Equation (43), and its associated component equations, were used togenerate the time series data for the photovoltaic fleet with theadditional specification of the specific empirical models, as describedin Equations (44) through (47). The resulting fleet power and change inpower is presented represented by the red lines in FIGS. 12E and 12F.

Probability Density Function

The conversion from area statistics to point statistics relied upon twoterms A_(Kt) and A_(ΔKt) to calculate σ_(Kt) ² and σ_(ΔKt) ²,respectively. This section considers these terms in more detail. Forsimplicity, the methodology supra applies to both Kt and ΔKt, so thisnotation is dropped. Understand that the correlation coefficient ρ^(i,j)could refer to either the correlation coefficient for clearness index orthe correlation coefficient for the change in clearness index, dependingupon context. Thus, the problem at hand is to evaluate the followingrelationship:

$\begin{matrix}{A = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (48)\end{matrix}$

The computational effort required to calculate the correlationcoefficient matrix can be substantial. For example, suppose that the onewants to evaluate variance of the sum of points within a 1 squarekilometer satellite region by breaking the region into one millionsquare meters (1,000 meters by 1,000 meters). The complete calculationof this matrix requires the examination of 1 trillion (10¹²) locationpair combinations.

Discrete Formulation

The calculation can be simplified using the observation that many of theterms in the correlation coefficient matrix are identical. For example,the covariance between any of the one million points and themselvesis 1. This observation can be used to show that, in the case of arectangular region that has dimension of H by W points (total of N) andthe capacity is equal distributed across all parts of the region that:

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right){\quad\left\lbrack {{\sum\limits_{i = 0}^{H - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {H - i} \right)\left( {W - j} \right)} \right\rbrack}\rho^{d}}}} + {\sum\limits_{i = 0}^{W - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {W - i} \right)\left( {H - j} \right)} \right\rbrack}\rho^{d}}}}} \right\rbrack}}} & (49)\end{matrix}$

where:

k=−1, when i=0 and j=0

-   -   1, when j=0 or j=i    -   2, when 0<j<i

When the region is a square, a further simplification can be made.

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right){\quad\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = 0}^{i}{2^{k}\left( {\sqrt{N} - i} \right)\left( {\sqrt{N} - j} \right)\rho^{d}}}} \right\rbrack}}} & (50)\end{matrix}$

where:

0, when  i = 0  and  j = 0k = 2, when  j = 0  or  j = i, and 3, when  0 < j < i$d = {\left( \sqrt{i^{2} + j^{2}} \right){\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right).}}$

The benefit of Equation (50) is that there are

$\frac{N - \sqrt{N}}{2}$

rather than unique combinations that need to be evaluated. In theexample above, rather than requiring one trillion possible combinations,the calculation is reduced to one-half million possible combinations.

Continuous Formulation

Even given this simplification, however, the problem is stillcomputationally daunting, especially if the computation needs to beperformed repeatedly in the time series. Therefore, the problem can berestated as a continuous formulation in which case a proposedcorrelation function may be used to simplify the calculation. The onlyvariable that changes in the correlation coefficient between any of thelocation pairs is the distance between the two locations; all othervariables are the same for a given calculation. As a result, Equation(50) can be interpreted as the combination of two factors: theprobability density function for a given distance occurring and thecorrelation coefficient at the specific distance.

Consider the probability density function. The actual probability of agiven distance between two pairs occurring was calculated for a 1,000meter×1,000 meter grid in one square meter increments. The evaluation ofone trillion location pair combination possibilities was evaluated usingEquation (48) and by eliminating the correlation coefficient from theequation. FIG. 14 is a graph depicting, by way of example, an actualprobability distribution for a given distance between two pairs oflocations, as calculated for a 1,000 meter×1,000 meter grid in onesquare meter increments.

The probability distribution suggests that a continuous approach can betaken, where the goal is to find the probability density function basedon the distance, such that the integral of the probability densityfunction times the correlation coefficient function equals:

A=∫f(D)p(d)dD  (51)

An analysis of the shape of the curve shown in FIG. 14 suggests that thedistribution can be approximated through the use of two probabilitydensity functions. The first probability density function is a quadraticfunction that is valid between 0 and √{square root over (Area)}.

$\begin{matrix}{f_{Quad} = \left\{ \begin{matrix}{\left( \frac{6}{Area} \right)\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)} & {{{for}\mspace{14mu} 0} \leq D \leq \sqrt{Area}} \\0 & {{{for}\mspace{14mu} D} > \sqrt{Area}}\end{matrix} \right.} & (52)\end{matrix}$

This function is a probability density function because integratingbetween 0 and √{square root over (Area)} equals 1, that is,P[0≤D≤√{square root over (Area)}]=∫₀^(√{square root over (Area)})f_(Quad)dD=1.

The second function is a normal distribution with a mean of √{squareroot over (Area)} and standard deviation of 0.1 √{square root over(Area)}.

$\begin{matrix}{f_{Norm} = {\left( \frac{1}{0.1*\sqrt{Area}} \right)\left( \frac{1}{\sqrt{2\pi}} \right)e^{{- {(\frac{1}{2})}}{(\frac{D - \sqrt{Area}}{0.1*\sqrt{Area}})}^{2}}}} & (53)\end{matrix}$

Likewise, integrating across all values equals 1.

To construct the desired probability density function, take, forinstance, 94 percent of the quadratic density function plus 6 of thenormal density function.

f=0.94∫₀ ^(√{square root over (Area)}) f _(Quad) dD+0.06∫_(−∞) ^(+∞) f_(Norm) dD  (54)

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

The result is that the correlation matrix of a square area with uniformpoint distribution as N gets large can be expressed as follows, firstdropping the subscript on the variance since this equation will work forboth Kt and ΔKt.

A≈[0.94∫₀ ^(√{square root over (Area)}) f _(Quad)ρ(D)dD+0.06∫_(−∞) ^(+∞)f _(Norm)ρ(D)dD]  (55)

where ρ(D) is a function that expresses the correlation coefficient as afunction of distance (D).

Area to Point Conversion Using Exponential Correlation Coefficient

Equation (55) simplifies the problem of calculating the correlationcoefficient and can be implemented numerically once the correlationcoefficient function is known. This section demonstrates how a closedform solution can be provided, if the functional form of the correlationcoefficient function is exponential.

Noting the empirical results as shown in the graph in FIGS. 9A-9F, anexponentially decaying function can be taken as a suitable form for thecorrelation coefficient function. Assume that the functional form ofcorrelation coefficient function equals:

$\begin{matrix}{{\rho (D)} = e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}} & (56)\end{matrix}$

Let Quad be the solution to ∫₀^(√{square root over (Area)})f_(Quad)·ρ(D) dD.

$\begin{matrix}{{Quad} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho (D)}{dD}}} = {\left( \frac{6}{Area} \right){\int_{0}^{\sqrt{Area}}{{\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)\left\lbrack e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}} \right\rbrack}{dD}}}}}} & (57)\end{matrix}$

Integrate to solve.

$\begin{matrix}{{Quad} = {(6)\left\lbrack {{\left( {{\frac{x}{\sqrt{Area}}D} - 1} \right)e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}} - {\left( {{\left( \frac{x}{\sqrt{Area}} \right)^{2}D^{2}} - {2\frac{x}{\sqrt{Area}}D} + 2} \right)e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}}} \right\rbrack}} & (58)\end{matrix}$

Complete the result by evaluating at D equal to √{square root over(Area)} for the upper bound and 0 for the lower bound. The result is:

$\begin{matrix}{{Quad} = {\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} & (59)\end{matrix}$

Next, consider the solution to ∫_(−∞) ^(+∞)f_(Norm)·ρ(D) dD, which willbe called Norm.

$\begin{matrix}{{Norm} = {\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}{(\frac{D - \mu}{\sigma})}^{2}}e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}{dD}}}}} & (60)\end{matrix}$

where μ=√{square root over (Area)} and σ=0.1√{square root over (Area)}.Simplifying:

$\begin{matrix}{{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack \left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{- {{(\frac{1}{2})}\lbrack\frac{D - {({\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}})}}{\sigma}\rbrack}^{2}}{dD}}}}}\mspace{20mu} {{{Substitute}\mspace{14mu} z} = {{\frac{D - \left( {\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}} \right)}{\sigma}\mspace{14mu} {and}\mspace{14mu} \sigma \; {dz}} = {{dD}.}}}} & (61) \\{\mspace{79mu} {{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}} \right\rbrack \left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}z^{2}}{dz}}}}}} & (62)\end{matrix}$

Integrate and solve.

$\begin{matrix}{{Norm} = e^{\frac{x}{\sqrt{Area}}{({\mu + {\frac{1}{2}\frac{x}{\sqrt{Area}}\sigma^{2}}})}}} & (63)\end{matrix}$

Substitute the mean of √{square root over (Area)} and the standarddeviation of 0.1√{square root over (Area)} into Equation (55).

Norm=e ^(x(1+0.005x))  (64)

Substitute the solutions for Quad and Norm back into Equation (55). Theresult is the ratio of the area variance to the average point variance.This ratio was referred to as A (with the appropriate subscripts andsuperscripts) supra.

$\begin{matrix}{A = {{0.94{\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} + {0.06e^{x{({1 + {0.005x}})}}}}} & (65)\end{matrix}$

Example

This section illustrates how to calculate A for the clearness index fora satellite pixel that covers a geographical surface area of 1 km by 1km (total area of 1,000,000 m²), using a 60-second time interval, and 6meter per second cloud speed. Equation (56) required that thecorrelation coefficient be of the form

$e^{\frac{x\mspace{11mu} D}{\sqrt{Area}}}.$

The empirically derived result in Equation (44) can be rearranged andthe appropriate substitutions made to show that the correlationcoefficient of the clearness index equals

${\exp\left\lbrack \frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)\mspace{11mu} D}{1000\mspace{14mu} {CloudSpeed}} \right\rbrack}.$

Multiply the exponent by

$\frac{\sqrt{Area}}{\sqrt{Area}},$

so that the correlation coefficient equals

$\exp {\left\{ {\left\lbrack \frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu} {CloudSpeed}} \right\rbrack\left\lbrack \frac{D}{\sqrt{Area}} \right\rbrack} \right\}.}$

This expression is now in the correct form to apply Equation (65), where

$x = {\frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu} {CloudSpeed}}.}$

Inserting the assumptions results in

${x = {\frac{\left( {{\ln \; 60} - 9.3} \right)\sqrt{1,000,000}}{1000 \times 6} = {- 0.86761}}},$

which is applied to Equation (65). The result is that A equals 65percent, that is, the variance of the clearness index of the satellitedata collected over a 1 km² region corresponds to 65 percent of thevariance when measured at a specific point. A similar approach can beused to show that the A equals 27 percent for the change in clearnessindex. FIG. 16 is a graph depicting, by way of example, resultsgenerated by application of Equation (65).

Time Lag Correlation Coefficient

This section presents an alternative approach to deriving the time lagcorrelation coefficient. The variance of the sum of the change in theclearness index equals:

σ_(ΣΔKt) ²=VAR[(Kt ^(Δt) −Kt)]  (66)

where the summation is over N locations. This value and thecorresponding subscripts have been excluded for purposes of notationalsimplicity. Divide the summation into two parts and add severalconstants to the equation:

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {{VAR}\left\lbrack {{\sigma_{\sum{Kt}^{\Delta \; t}}\left( \frac{\sum{Kt}^{\Delta \; t}}{\sigma_{\sum{Kt}^{\Delta \; t}}} \right)} - {\sigma_{\sum{Kt}}\left( \frac{\sum{Kt}}{\sigma_{\sum{Kt}}} \right)}} \right\rbrack}} & (67)\end{matrix}$

Since σ_(ΣKt) _(Δt) ≈σ_(ΣKt) (or σ_(ΣKt) _(Δt) =σ_(ΣKt) if the firstterm in Kt and the last term in Kt^(Δt) are the same):

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}{{VAR}\left\lbrack {\frac{\sum{Kt}^{\Delta \; t}}{\sigma_{\sum{Kt}^{\Delta \; t}}} - \frac{\sum{Kt}}{\sigma_{\sum{Kt}}}} \right\rbrack}}} & (68)\end{matrix}$

The variance term can be expanded as follows:

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}\left\{ {\frac{{VAR}\left\lbrack {\sum{Kt}^{\Delta \; t}} \right\rbrack}{\sigma_{\sum{Kt}^{\Delta \; t}}^{2}} + \frac{{VAR}\left\lbrack {\sum{Kt}} \right\rbrack}{\sigma_{\sum{Kt}}^{2}} - \frac{2{{COV}\left\lbrack {{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} \right\rbrack}}{\sigma_{\sum{Kt}}\sigma_{\sum{Kt}^{\Delta \; t}}}} \right\}}} & (69)\end{matrix}$

Since COV[ΣKt,ΣKt^(Δt)]=σ_(ΣKt)σ_(Kt) _(Δt) ρ^(ΣKt,ΣKt) ^(Δt) , thefirst two terms equal one and the covariance term is replaced by thecorrelation coefficient.

σ_(ΣΔKt) ²=2σ_(ΣKt) ²(1−ρ^(ΣKt,ΣKt) ^(Δt) )  (70)

This expression rearranges to:

$\begin{matrix}{\rho^{{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} = {1 - {\frac{1}{2}\frac{\sigma_{\sum{\Delta \; {Kt}}}^{2}}{\sigma_{\sum{Kt}}^{2}}}}} & (71)\end{matrix}$

Assume that all photovoltaic plant ratings, orientations, and areaadjustments equal to one, calculate statistics for the clearness aloneusing the equations described supra and then substitute. The result is:

$\begin{matrix}{\rho^{{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} = {1 - \frac{P^{\Delta \; {Kt}}\sigma \frac{2}{\Delta \; {Kt}}}{2P^{Kt}\sigma \frac{2}{Kt}}}} & (72)\end{matrix}$

Relationship Between Time Lag Correlation Coefficient and Power/Changein Power Correlation Coefficient

This section derives the relationship between the time lag correlationcoefficient and the correlation between the series and the change in theseries for a single location.

$\rho^{P,{\Delta \; P}} = {\frac{{COV}\left\lbrack {P,{\Delta \; P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}} = {\frac{{COV}\left\lbrack {P,{P^{\Delta \; t} - P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}} = \frac{{{COV}\left\lbrack {P,P^{\Delta \; t}} \right\rbrack} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}}}}$Since  σ_(Δ P)² = VAR[P^(Δ t) − P] = σ_(P)² + σ_(P^(Δ t))² − 2 COV[P, P^(Δ t)]   and${{{COV}\left\lbrack {P,P^{\Delta \; t}} \right\rbrack}\; = {\rho^{P,P^{\Delta \; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta \; t}}^{2}}}},{then}$$\rho^{P,{\Delta \; P}} = {\frac{{\rho^{P,P^{\Delta \; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta \; t}}^{2}}} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\left( {\sigma_{P}^{2} + \sigma_{P^{\Delta \; t}}^{2} - {2\rho^{P,{P\; \Delta \; t}}\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}}} \right)}}.}$

Since σ_(P) ²≈_(P) _(Δt) , this expression can be further simplified.Then, square both expression and solve for the time lag correlationcoefficient:

ρ^(P,P) ^(Δt) =1−2(ρ^(P,ΔP))²

Correlation Coefficients Between Two Regions

Assume that the two regions are squares of the same size, each side withN points, that is, a matrix with dimensions of √{square root over (N)}by √{square root over (N)} points, where √{square root over (N)} is aninteger, but are separated by one or more regions. Thus:

$\begin{matrix}{{{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( \frac{1}{N^{2}} \right)\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = {1 - \sqrt{N}}}^{\sqrt{N} - 1}{{k\left( {\sqrt{N} - i} \right)}\left( {\sqrt{N} - {j}} \right)\rho^{d}}}} \right\rbrack}}\mspace{20mu} {{{where}\mspace{14mu} k} = \left\{ {\begin{matrix}1 & {{{when}\mspace{14mu} i} = 0} \\2 & {{{when}\mspace{14mu} i} > 0}\end{matrix},\mspace{20mu} {d = {\left( \sqrt{i^{2} + \left( {j + {M\sqrt{N}}} \right)^{2}} \right)\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right)}},} \right.}} & (73)\end{matrix}$

and M equals the number of regions.

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions. FIG. 17suggests that the probability density function can be estimated usingthe following distribution:

$\begin{matrix}{f = \left\{ \begin{matrix}{1 - \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{{for}{\mspace{11mu} \;}{Spacing}} - \sqrt{Area}} \leq D \leq {Spacing}} \\{1 + \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{for}{\mspace{11mu} \;}{Spacing}} \leq D \leq {{Spacing} + \sqrt{Area}}} \\0 & {{all}\mspace{14mu} {else}}\end{matrix} \right.} & (74)\end{matrix}$

This function is a probability density function because the integrationover all possible values equals zero. FIG. 18 is a graph depicting, byway of example, results by application of this model.

Inferring Photovoltaic System Configuration Specifications

Accurate power output forecasting through photovoltaic power predictionmodels, such as described supra, requires equally precise solarirradiance data and photovoltaic system configuration specifications.Solar irradiance data can be obtained from ground-based measurements,satellite imagery, numerical weather prediction models, as well asthrough various reliable third party sources, such as the Solar Anywhereservice (http://www.SolarAnywhere.com), a Web-based service operated byClean Power Research, L.L.C., Napa, Calif., that can providesatellite-derived solar irradiance data forecasted up to seven daysahead of time and archival solar irradiance data, dating back to Jan. 1,1998, at time resolutions of as fast as one minute for historical dataup to several hours forecasted and then transitioning to a one-hour timeresolution up to seven days ahead of time.

On the other hand, obtaining accurate and reliable photovoltaic plantconfiguration specifications for individual photovoltaic systems can bea challenge, particularly when the photovoltaic systems are part of ageographically dispersed power generation fleet. Part of the concernarises due to an increasing number of grid-connected photovoltaicsystems that are privately-owned residential and commercial systems,where they are neither controlled nor accessible by grid operators andpower utilities, who require precise configuration specifications forplanning and operations purposes or whether they are privately-ownedutility-scale systems for which specifications are unavailable.Moreover, in some situations, the configuration specifications may beeither incorrect, incomplete or simply not available.

Photovoltaic plant configuration specifications can be accuratelyinferred through analysis of historical measurements of the photovoltaicplant's production data and measured historical irradiance data. FIG. 19is a flow diagram showing a computer-implemented method 180 forinferring operational specifications of a photovoltaic power generationsystem 25 (shown in FIG. 2) in accordance with a further embodiment.Configuration data include the plant's power rating and electricalcharacteristics, including the effect of the efficiency of the modules,wiring, inverter, and other factors; and operational features, includingtracking mode (fixed, single-axis tracking, dual-axis tracking), azimuthangle, tilt angle, row-to-row spacing, tracking rotation limit, andshading or other physical obstructions. Shading and physicalobstructions can be evaluated by specifying obstructions as part of aconfiguration. For instance, an obstruction could be initially definedat an azimuth angle between 265° and 275° with a 10° elevation (tilt)angle. Additional configurations would vary the azimuth and elevationangles by fixed amounts, thereby exercising the range of possibleobstruction scenarios. The method 180 can be implemented in software andexecution of the software can be performed on a computer system 21, suchas described supra with reference to FIG. 2, as a series of process ormethod modules or steps.

Configuration specifications can be inferred through evaluation ofmeasured historical photovoltaic system production data and measuredhistorical resource data. First, measured historical time-seriesphotovoltaic system production data and geographical coordinates arerespectively obtained for the photovoltaic power generation system 25under evaluation (steps 181 and 182). Optionally, the production datacan be segmented into multiple time periods for calculating the system'spower rating during different times of the year (step 183). A set ofphotovoltaic plant configuration specifications is then inferred foreach of the time periods, if applicable (steps 184-194), as follows.First, based on the measured historical production data, the output of anormalized 1-kW-AC photovoltaic system is simulated for the current timeperiod for a wide range of hypothetical (or model) photovoltaic systemconfigurations (step 186), as further described infra with reference toFIG. 20.

Following simulation, each of the hypothetical photovoltaic systemconfigurations is evaluated (steps 186-191), as follows. The totalmeasured energy produced over the selected time period (excluding anytimes with erroneous measured data, which are screened out duringsimulation, as explained infra) is determined (step 187). The ratio ofthe total measured energy over the total simulated energy is calculated(step 188), which produces a simulated photovoltaic system rating.However, system power ratings other than the ratio ofmeasured-to-simulated energy could be used.

Assuming that a photovoltaic simulation model that scales linearly (ornear-linearly, that is, approximately or substantially linear, such asdescribed infra beginning with reference to Equation (12)) inphotovoltaic system rating was used, each point in the simulated timeseries of power production data is then proportionately scaled up by thesimulated photovoltaic system rating (step 189). Each of the points inthe simulated and measured time series of power production data arematched up and the error between the measured and simulated power outputis calculated (step 190) using standard statistical methodologies. Forexample, the relative mean absolute error (rMAE) can be used, such asdescribed in Hoff et al., “Modeling PV Fleet Output Variability,” SolarEnergy 86, pp. 2177-2189 (2012) and Hoff et al, “Reporting of IrradianceModeling Relative Prediction Errors,” Progress in Photovoltaics: Res.Appl. DOI: 10.1002/pip.2225 (2012) the disclosure of which isincorporated by reference. Other methodologies, including but notlimited to root mean square error, to calculate the error between themeasured and simulated data could also be used. Each hypotheticalphotovoltaic system configuration is similarly evaluated (step 191).

Once all of the configurations have been explored (steps 186-191), avariance threshold is established and the variance between the measuredand simulated power outputs of all the configurations is taken (step192) to ensure that invalid data has been excluded. The hypotheticalphotovoltaic system configuration, including, but not limited to,tracking mode (fixed, single-axis tracking, dual-axis tracking), azimuthangle, tilt angle, row-to-row spacing, tracking rotation limit, andshading configuration, that minimizes error is selected (step 193). Theselected configuration represents the inferred photovoltaic systemconfiguration specification for the photovoltaic power generation system25 under evaluation for the current time period. Each time period issimilarly evaluated (step 194). Once all of the time periods have beenexplored (steps 184-194), an inferred photovoltaic system configurationspecification will have been selected for each time period. Ideally, thesame configuration will have been selected across all of the timeperiods. However, in the event of different configurations having beenselected, the configuration with the lowest overall error (step 193) canbe picked. Alternatively, other tie-breaking configuration selectioncriteria could be applied, such as the system configurationcorresponding to the most recent set of production data. In a furtherembodiment, mismatched configurations from each of the time periods mayindicate a concern outside the scope of plant configuration evaluation.For instance, the capacity of a plant may have increased, therebyenabling the plant to generate more power that would be reflected by asimulation based on the hypothetical photovoltaic system configurationswhich were applied. (In this situation, the hypothetical photovoltaicsystem configurations would have to be modified beginning at the timeperiod corresponding to the supposed capacity increase.) Still othertie-breaking configuration selection criteria are possible.

In addition, the range of hypothetical (or model) photovoltaic systemconfigurations used in inferring the system's “optimal” configurationdata, that is, a system configuration heuristically derived throughevaluation of different permutations of configuration parameters,including power rating, electrical characteristics, and operationalfeatures, can be used to look at the affect of changing theconfiguration in view of historical measured performance. For instance,while the hypothetical configuration that minimizes error signifies theclosest (statistical) fit between actual versus simulated powergeneration models, other hypothetical configurations may offer thepotential to improve performance through changes to the plant'soperational features, such as revising tracking mode (fixed, single-axistracking, dual-axis tracking), azimuth, tilt, row-to-row spacing,tracking rotation limit, and shading configurations. Moreover, theaccuracy or degree to which a system configuration is “optimal” can beimproved further by increasing the degree by which each of theconfiguration parameters is varied. For instance, tilt angle can bepermuted in one degree increments, rather than five degrees at a time.Still other ways of structuring or permuting the configurationparameters, as well as other uses of the hypothetical photovoltaicsystem configurations, are possible.

Optionally, the selected photovoltaic system configuration can be tuned(step 195), as further described infra with reference to FIG. 24. Theselected and, if applicable, tuned photovoltaic system configuration isthen provided (step 196) as the inferred photovoltaic systemconfiguration specifications, which can be used to correct, replace or,if configuration data is unavailable, stand-in for the system'sspecifications.

Power Output Simulation

Photovoltaic power prediction models are typically used in forecastingpower generation, but prediction models can also be used to simulatepower output for hypothetical photovoltaic system configurations. Thesimulation results can then be evaluated against actual historicalmeasured photovoltaic production data and statistically analyzed toidentify the inferred (and most probable) photovoltaic systemconfiguration specification. FIG. 20 is a flow diagram showing a routine200 for simulating power output of a photovoltaic power generationsystem 25 for use in the method 180 of FIG. 19. Power output issimulated for a wide range of hypothetical photovoltaic systemconfigurations, which are defined to exercise the different types ofphotovoltaic system configurations possible. Each of the hypotheticalconfigurations may vary based on power rating and electricalcharacteristics, including the effect of the efficiency of the solarmodules, wiring, inverter, and related factors, and by their operationalfeatures, such as size and number of photovoltaic arrays, the use offixed or tracking arrays, whether the arrays are tilted at differentangles of elevation or are oriented along differing azimuthal angles,and the degree to which each system is covered by shade on a row-to-rowbasis or due to cloud cover or other physical obstructions. Still otherconfiguration details are possible.

Initially, historical measured irradiance data for the current timeperiod is obtained (step 201), such as described supra beginning withreference to FIG. 3. Preferably, the irradiance data includes isobtained from a solar resource data set that contains both historicaland forecasted data, which allows further comparative analysis. Each ofthe hypothetical photovoltaic system configurations are evaluated (steps202-206), as follows. Optionally, the measured irradiance data isscreened (step 203) to eliminate data where observations are invalideither due to data recording issues or photovoltaic system performanceissues power output. The production data, that is, measured poweroutput, is correspondingly updated (step 204). Finally, power output issimulated based on the current system configuration and the measuredirradiance data for the current time period (step 205), such asdescribed supra beginning with reference to Equation (12). In oneembodiment, a normalized 1-kW-AC photovoltaic system is simulated, whichfacilitates proportionately scaling the simulated power output based onthe ratio (or function) of measured-to-simulated energy. A differentapproach may be required for photovoltaic simulation models that do notscale linearly (or near-linearly) with system rating. For instance, anon-linear (or non-near-linear) simulation model may need to be runmultiple times until the system rating for the particular systemconfiguration results in the same annual energy production as themeasured data over the same time period. Still other approaches toscaling non-linear (or non-near-linear) simulation model results toactual measured energy output are possible. Each system configuration issimilarly evaluated (step 206), after which power production simulationfor the current time period is complete.

Example of Inferred Photovoltaic Plant Configuration Specifications

The derivation of a simulated photovoltaic system configuration can beillustrated with a simple example. FIG. 21 is a table showing, by way ofexample, simulated half-hour photovoltaic energy production for a1-kW-AC photovoltaic system. Each column represents a differenthypothetical photovoltaic system configuration. For instance, the firstcolumn represents a horizontal photovoltaic plant with a fixed array ofsolar panels set at a 180 degree azimuth with zero tilt. Each rowrepresents the power produced at each half-hour period for a 1-kW-ACphotovoltaic system, beginning on Jan. 1, 2012 (night time half-hourperiods, when solar power production is zero, are omitted for clarity).The simulated energy production data covers the time period from Jan. 1,2012 through Dec. 31, 2012, although only the first few hours of Jan. 1,2012 are presented in FIG. 21. The latitude and longitude of thephotovoltaic system were obtained and the Solar Anywhere service, citedsupra, was used to obtain both historical and forecasted solar data andto simulate photovoltaic power output generation.

The simulated energy production can be compared to actual historicaldata. Here, in 2012, the photovoltaic plant produced 12,901,000 kWh intotal measured energy, while the hypothetical photovoltaic systemconfiguration represented in the first column had a simulated output of1,960 kWh over the same time period (for a 1-kW-AC photovoltaic system).Assuming that a linearly-scalable (or near-linearly scalable)photovoltaic simulation model was used, the simulated output of 1,960kWh implies that this particular system configuration would need arating of 6,582 kW-AC to produce the same amount of energy, that is,12,901,000 kWh, as the actual system. Thus, each half hour value can bemultiplied by 6,582 to match simulated to actual power output.

The results can be visually presented. FIG. 22 are graphs depicting, byway of example, simulated versus measured power output 12 forhypothetical photovoltaic system configuration specifications evaluatedusing the method 180 of FIG. 19. Each of the graphs corresponds tophotovoltaic power as produced under a different hypotheticalphotovoltaic system configuration, as shown in the columns of the tableof FIG. 21. The x-axis of each graph represents measured power output inmegawatts (MW). The y-axis of each graph represents simulated poweroutput in megawatts (MW). Within each graph, the points present thehalf-hour simulated versus measured photovoltaic power data. Visually,the simulated versus measured power output data for the fixed systemconfiguration with a 180 degree azimuth angle and 15 degree tilt showsthe closest correlation.

Similarly, FIG. 23 is a graph depicting, by way of example, the rMAEbetween the measured and simulated power output for all systemconfigurations as shown in FIG. 22. The x-axis represents the percentageof rMAE for half-hour intervals. The y-axis represents the differenthypothetical photovoltaic system configurations. Again, the fixed systemconfiguration with a 180 degree azimuth angle and 15 degree tiltreflects the lowest rMAE and accordingly provides an optimal systemconfiguration.

Optimizing Photovoltaic System Configuration Specifications

Truly perfect weather data does not exist, as there will always beinaccuracies in weather data, whether the result of calibration or othererrors or incorrect model translation. In addition, photovoltaic plantperformance is ultimately unpredictable due to unforeseeable events andcustomer maintenance needs. For example, a power inverter outage is anunpredictable photovoltaic performance event, while photovoltaic panelwashing after a long period of soiling is an example of an unpredictablecustomer maintenance event.

In a further embodiment, the power calibration model can be tuned toimprove forecasts of power output of the photovoltaic plant based on theinferred (and optimal) photovoltaic plant configuration specification,such as described in commonly-assigned U.S. Patent application, entitled“Computer-Implemented Method for Tuning Photovoltaic Power GenerationPlant Forecasting,” cited supra. Tuning helps to account for subtletiesnot incorporated into the selected photovoltaic simulation model,including any non-linear (or non-near-linear) issues in the power model.FIG. 24 are graphs depicting, by way of example, simulated versusmeasured power output for the optimal photovoltaic system configurationspecifications as shown in FIG. 22. The graphs corresponds tophotovoltaic power as produced under the optimal photovoltaic systemconfiguration, that is, a fixed system configuration with a 180 degreeazimuth angle and 15 degree tilt before (left graph) and after (rightgraph) tuning. The x-axis of each graph represents measured power outputin megawatts (MW). The y-axis of each graph represents simulated poweroutput in megawatts (MW).

Referring first to the “before” graph, the simulated power productiondata over-predicts power output during lower measured power conditionsand under-predicts power output during high measured power conditions.Referring next to the “after” graph, tuning removes the uncertaintyprimarily due to irradiance data error and power conversioninaccuracies. As a result, the rMAE (not shown) is reduced from 11.4percent to 9.1 percent while also eliminating much of the remainingbias.

Inferring Photovoltaic System Configuration Specifications Using NetLoad Data

The historical measured photovoltaic power production data, which isnecessary for inferring operational photovoltaic system configurationspecifications per the approach described supra beginning with referenceto FIG. 19, may not always be available. High-quality, historical,measured photovoltaic production data may be unavailable for severalreasons. For example, photovoltaic system production may not be directlymonitored. Alternatively, photovoltaic system production may bemonitored by one party, such as the photovoltaic system owner, but theproduction data may be unavailable to other interested parties, such asan electric utility. Moreover, even where a photovoltaic systemproduction is monitored and production data is available, the quality ofthe production data may be questionable, unreliable, or otherwiseunusable. These situations are particularly prevalent with photovoltaicsystems located on the premises of residential utility customers.

In a further embodiment, net load data for a building can be used toinfer operational photovoltaic system configuration specifications, asan alternative to historical measured photovoltaic system productiondata. Smart electric meters provide one source of net load data. Smartmeters are becoming increasingly commonplace, as power utilities movetowards tiered and time-of-use electricity pricing structures, whichrequire knowledge of when and how much power is consumed based on thetime of day and season. Smart meters also allow a power utility tomonitor the net power load of a building, but generally not the powerloads of individual appliances or machinery (collectively,“components”).

Net Load Characterization

Smart meters typically record detailed time series data for individualcustomers. In most cases, the various component loads are not directlymeasureable; component load measurement would require the smart meter tobe able to identify when specific components began and ceased operation,which is largely impracticable. As a result, only net load is availableand individual component loads must be estimated.

Assuming that a building has only one point of electricity service, netelectricity load during any given time interval, such as measured by asmart meter, equals the sum of multiple component loads, minus anyon-site distributed generation. For the sake of discussion, aphotovoltaic system will be assumed to provide all on-site distributedgeneration, although other sources of on-site distributed generation arepossible. Only one photovoltaic system is necessary, because the outputfrom any individual photovoltaic systems situated in the same locationwould be correlated. Thus, the photovoltaic system performance can bespecified by a single operational photovoltaic system configuration.

Individual component loads represent the load associated with groups ofdevices with similar load characteristics. For example, all lights onthe same circuit are associated with a single component load because thelight work in tandem with each other.

Component loads can be characterized into three types. A Base Loadrepresents constant power that is drawn at all times. A Binary Loadrepresents a load that is either on or off, and which, when on, drawspower at a single relatively stable power level. For example, the powerdrawn by a refrigerator is a binary load. Finally, a Variable Loadrepresents a load that can take on multiple power levels. For instance,the power drawn by an electric range is variable, as the load depends onthe number of stove burners in use and their settings.

In any given building, there is one base load, one or more binary loads,and one or more variable loads. The base load equals the sum of allcomponent loads that are on at all times. The binary loads equal the sumof the all component loads that are binary during the time that thecomponents are on. Each binary load can be expressed as the product ofan indicator function, that is, a value that is either 0 or 1, and abinary load level (energy consumption) for that binary load. Whenmultiplied by the binary load level, the indicator function acts as anidentify function that returns the binary load level only when the valueof the indicator function is 1. When the value of the indicator functionis 0, the binary load level is 0, which masks out the binary load. Thevariable loads equal the sum of all component loads that are variableduring the time that the components are on based on their correspondingvariable load levels (energy consumption).

A Net Load at time interval t can be expressed as:

$\begin{matrix}{{{Net}\mspace{14mu} {Load}_{t}} = {{{Base}\mspace{14mu} {Load}} + {\sum\limits_{m = 1}^{M}{1_{t}^{m} \times {Binary}\mspace{14mu} {Load}^{m}}} + {\sum\limits_{n = 1}^{N}{{Variable}\mspace{14mu} {Load}_{t}^{n}}} - {PV}_{t}}} & (75)\end{matrix}$

where Base Load represents the base load, M is the number of binaryloads, Binary Load^(m) represents component binary load m, 1_(t) ^(m) isan indicator function at time interval t, N is the number of variableloads, Variable Load_(t) ^(n) represents component variable load m attime interval t. There is no time subscript on Base Load because thebase load is the same at all times. The indicator function (1_(t) ^(m))is either 0 or 1 for component Binary Load^(m) at time interval t. Thevalue is 0 when the load is off, and the value is 1 when the load on.There is a time subscript (t) on the indicator function, but there is notime subscript on the Binary Load proper because the binary load isconstant when on.

Photovoltaic Production

Photovoltaic production can be solved for by rearranging Equation (75)if the net load and all individual load components are known at a giventime interval. Photovoltaic production PV_(t) at time interval t can berepresented by the normalized photovoltaic production for a 1-kWphotovoltaic system for a particular configuration times the rating ofthe system, such that:

PV_(t)=(Rating)(PV _(t))  (76)

where Rating is the rating of the photovoltaic system in kilowatts (kW),and PV_(t) is the production associated with a normalized 1-kWphotovoltaic system for a particular photovoltaic system configuration.

Given an accurate photovoltaic simulation model, the normalizedphotovoltaic production PV _(t) at time interval t can be expressed as afunction of photovoltaic system configuration and solar resource data.The photovoltaic system configuration is not dependent on time, whilethe solar resource data is dependent on time:

PV _(t) =f(Config,Solar_(t))  (77)

where Config represents a set of photovoltaic system configurationparameters, for instance, azimuth, tilt, tracking mode, and shading, fora normalized 1-kW photovoltaic system and Solar is the solar resourceand other weather data, including normalized horizontal irradiation,average ambient temperature, and wind speed, at time interval t.

Substitute Equation (77) into Equation (76),then into Equation (75):

$\begin{matrix}{{{Net}\mspace{14mu} {Load}_{t}} = {{{Base}\mspace{14mu} {Load}} + {\sum\limits_{m = 1}^{M}{1_{t}^{m} \times {Binary}\mspace{14mu} {Load}^{m}}} + {\sum\limits_{n = 1}^{N}{{Variable}\mspace{14mu} {Load}_{t}^{n}}} - {({Rating}) \times {f\left( {{Config},{Solar}_{t}} \right)}}}} & (78)\end{matrix}$

Estimate Component Loads

In most cases, the various component loads, that is, the binary loadsand the variable loads, are not directly measureable and only the netload is available. As a result, the component loads must be estimated.

Simple Case

In the simplest scenario, there is only a base load and the net loadwill directly correspond to the base load. Adding a binary componentload complicates the simplest scenario. For purposes of illustrations,assume one binary load Binary Load* and Equation (78) simplifies to:

Net Load_(t)=Base Load+1_(t)*×Binary Load*(Rating)×f(Config.,Solar_(t))  (79)

Consider the analysis over a 24-hour period using a one-hour timeinterval. Solving Equation (79) yields an array of 24 net load values:

$\begin{matrix}{\begin{bmatrix}{{Net}\mspace{14mu} {Load}_{1}} \\\ldots \\{{Net}\mspace{14mu} {Load}_{24}}\end{bmatrix} = {{{Base}\mspace{14mu} {{Load}{\mspace{11mu} \;}\begin{bmatrix}1 \\\ldots \\1\end{bmatrix}}} + {{Binary}\mspace{14mu} {{Load}^{*}\;\begin{bmatrix}1_{1}^{*} \\\ldots \\1_{24}^{*}\end{bmatrix}}} - {({Rating})\begin{bmatrix}{f\left( {{{Config}.},{Solar}_{1}} \right)} \\\ldots \\{f\left( {{{Config}.},{Solar}_{24}} \right)}\end{bmatrix}}}} & (80)\end{matrix}$

where Net Load_(t) is the net load at time interval t. The system of 24equations expressed by Equation (80) has many unique variables and isdifficult to solve.

When the variables on the right-hand side of Equation (80) areparameterized into a set of key parameters, Equation (80) can be used toestimate net load for each time period. The key parameters include theBase Load, any Binary Loads, an Variable Loads (not shown in Equation(80)), photovoltaic system configurations (Config), and solar resourceand other weather data (Solar). The photovoltaic system ratings includepower ratings hypothesized for the plant for which a net load is beingestimated. Other key parameters are possible.

Simplifying Assumptions

This solution space of Equation (80) can be reduced in several ways.First, the photovoltaic production values are not 24 unrelated hourlyvalues. Rather, the values are related based on photovoltaic systemconfiguration and weather data input. Given accurate weather data, the24 photovoltaic values reduce to only one unknown variable, which issystem orientation. Second, a particularly interesting type of binaryload is a temperature-related binary load, which is related to the timeof day. The indicator function (1_(t)*) for a time-related binary loadequals 1 when the hours are between h₁ and h₂, and 0 for all otherhours. As a result, rather than requiring 24 values, only h₁ and h₂ arerequired to find the binary load.

Thus, Equation (80) can be simplified to require only two types of data,historical time series data, as expressed by net load and solar resourcedata, and a set of unknown parameters, which include photovoltaic systemrating, photovoltaic system configuration, base load, binary load,binary load indicator function start hour h₁, and binary load indicatorfunction end hour h₂.

Net loads are typically measured by a power utility on an hourly basis,although other net load measurement intervals are possible. The solarirradiance data, as well as simulation tools, can be provided by thirdparty sources, such as the SolarAnywhere data grid web interface, which,by default, reports irradiance data for a desired location using asingle observation time, and the SolarAnywhere photovoltaic systemmodeling service, available in the SolarAnywhere Toolkit, that useshourly resource data and user-defined physical system attributes tosimulate configuration-specific photovoltaic system output.SolarAnywhere is available online (http://www.SolarAnywhere.com) throughWeb-based services operated by Clean Power Research, L.L.C., Napa,Calif. Other sources of the solar irradiance data are possible,including numeric weather prediction models.

Total Squared Error

Let Net Load_(t) ^(Estimated) represent the estimated net load at timeinterval t based on the key parameters input to the right-hand side ofEquation (80), as described supra. The total squared error associatedwith the estimation equals:

$\begin{matrix}{{{Total}{\mspace{11mu} \;}{Squared}{\mspace{11mu} \;}{Error}} = {\sum\limits_{t = 1}^{24}\left( {{{Net}\mspace{14mu} {Load}_{t}} - {{Net}\mspace{14mu} {Load}_{1}^{Estimated}}} \right)^{2}}} & (81)\end{matrix}$

The key parameters should be selected to minimize the Total SquaredError, using a minimization approach, such as described supra withreference to FIG. 19.

Method

Photovoltaic plant configuration specifications can be accuratelyinferred with net load data applied to minimize total squared error.FIG. 25 is a flow diagram showing a computer-implemented method 220 forinferring operational specifications of a photovoltaic power generationsystem 25 (shown in FIG. 2) using net load data in accordance with afurther embodiment. The method 180 can be implemented in software andexecution of the software can be performed on a computer system 21, suchas described supra with reference to FIG. 2, as a series of process ormethod modules or steps.

As a preliminary step, time series net load data is obtained (step 221),which could be supplied, for instance, by a smart meter that monitor thenet power load of a building. Other source of net load data arepossible. An appropriate time period is then selected (step 222).Preferably, a time period with minimum or consistent power consumptionis selected. Longer duration, possibly contiguous time periods providebetter results, than shorter duration, temporally-distinct time periods.For residential applications, such time periods correspond to when theoccupants are on vacation or away from home for an extended period oftime. For commercial applications, such time periods correspond to aweekend or holiday when employees are away from work. Still otherappropriate time periods are possible.

Next, based on the historical solar resource and other weather data, theoutput of a normalized 1-kW-AC photovoltaic system is simulated for thecurrent time period for a wide range of hypothetical (or model)photovoltaic system configurations (step 223), as further describedsupra with reference to FIG. 20. Power generation data is simulated fora range of hypothetical photovoltaic system configurations based on anormalized solar power simulation model. Net load data is estimatedbased on a base load and, if applicable, any binary loads and anyvariable loads net load is estimated (step 224) by selecting keyparameters, per Equation (80). The key parameters include the base load,any binary loads, any variable loads, photovoltaic systemconfigurations, and solar resource and other weather data. Thephotovoltaic system ratings include power ratings hypothesized for theplant. Other key parameters are possible. As explained supra, a specialcase exists when there is only one binary load that is bothtemperature-related and only occurring between certain hours of the day.As well, a special case exists when there are no variable loads.

Finally, total squared error between the estimated and actual net loadfor each time period is minimized (step 225). The set of key parameterscorresponding to the net load estimate that minimizes the total squarederror with the measured net load data, per Equation (81), represents theinferred specifications of the photovoltaic plant configuration.

Photovoltaic system configurations are included as one of the keyparameters. A set of hypothetical photovoltaic system configurations aredefined that include power ratings and operational features, including,but not limited to, tracking mode (fixed, single-axis tracking,dual-axis tracking), azimuth angle, tilt angle, row-to-row spacing,tracking rotation limit, and shading configuration. The selectedconfiguration represents the inferred photovoltaic system configurationspecification for the photovoltaic power generation system 25 underevaluation for the current time period. In turn, the photovoltaic systemconfiguration that is part of the set of key parameters that minimizethe total squared error will become the inferred system specification.

Results

By way of example, consider inferring a photovoltaic plant configurationspecification for a house with a photovoltaic system using measured netload data. This scenario represents a good test for a variety ofreasons. First, the photovoltaic system is relatively small (1.7 kW whennew) and thus its effect is more difficult to measure relative to load.In addition, the photovoltaic system has been operating for more than adecade and has experienced significant degradation. As well, bothmeasured monthly photovoltaic production data and detailed photovoltaicsystem specifications exist. The first two reasons make this example agood test, and the third and fourth reasons can be used to validateaccuracy.

Obtain Time Series Net Load Data

The first step is to obtain time series net load data. Hourly,historical net load data was obtained for a one-year period for aresidential customer in California.

Select Time Period with Minimum and/or Consistent Consumption

The time series net load data was then evaluated to identify a period ofconsistent and minimum consumption. Daily power consumption wascalculated and a period of consecutive days with the lowest powerconsumption was identified. FIG. 26 is a graph depicting, by way ofexample, power consumption by the exemplary house over a one-yearperiod. The x-axis represents months, running from Oct. 12, 2012 throughOct. 13, 2013. They-axis represents power consumption in kWh per day.The time period with the most consistent and minimum consumptionoccurred between August 8 and Aug. 14, 2013 (indicated by thedownward-facing arrow), which corresponded to when the family was onvacation. FIG. 27 is a graph depicting, by way of example, net load datafor the exemplary house for a one-week period. The x-axis representsdays, running from Aug. 8, 2013 through Aug. 14, 2013. The y-axisrepresents power consumption in kWh. The time series net load data isindicated by the line of Smart Meter Data.

Estimate Net Load by Selecting Key Parameters with Goal of MinimizingError

The next step is to estimate net load by selecting key parameters withthe goal of minimizing total squared error. FIG. 28 is a graphdepicting, by way of example, measured net load data minus estimatedbase load data for the exemplary house for the one-week period. FIG. 29is a graph depicting, by way of example, measured net load data minusestimated base load data and estimated temperature-based attic fan loaddata for the exemplary house for the one-week period. The x-axesrepresents days, running from Aug. 8, 2013 through Aug. 14, 2013. They-axis represents power consumption in kWh. Referring to FIG. 29, theamount of net load remaining represents photovoltaic production. FIG. 30is a graph depicting, by way of example, implied photovoltaic productioncompared to the simulated photovoltaic production for the exemplaryhouse for the one-week period. The x-axes represents days, running fromAug. 8, 2013 through Aug. 14, 2013. The y-axis represents powerproduction in kWh. The result of the optimization is that the best fitphotovoltaic system is an east-facing, 22°-tilted photovoltaic systemwith a rating of 0.614 kW-DC. This system was estimated to have producedan average of 2.98 kWh per day during the 6-day period. FIGS. 31A-F aregraphs depicting, by way of example, comparing measured and simulatednet photovoltaic power production. The x-axes represents days, runningfrom Aug. 8, 2013 through Aug. 14, 2013. The y-axis represents powerconsumption in kWh.

Validation

The accuracy of the result that the photovoltaic system was estimated toproduce 2.98 kWh per day during the 6-day period can be validated usingmeasured data. The photovoltaic system in the foregoing example has beenin operation for more than 10 years. Monthly metering readings of thesystem production have been recorded for the past four years. FIG. 32 isa graph depicting, by way of example, photovoltaic production for afour-year period. The x-axis represents years, beginning at January 2010and ending at January 2014. The y-axis represents power production inkWh per day. The photovoltaic system actually produced 2.98 kWh per dayon average during the month of August, 2013 (as indicated by the blackcircle), which is essentially identical to what the foregoingmethodology predicted.

Note that the measured power production (line 322) was significantlyless than expected, as can be seen by comparison to expected powerproduction (line 321). This photovoltaic system is either in need ofmaintenance or cleaning (the last cleaning occurred in August, 2010), orhas experienced significant power production degradation.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A system net load-based inference of operationalspecifications of a photovoltaic power generation system with the aid ofa digital computer, comprising: a data storage comprisingcomputer-readable data, comprising: a time series of net load data forpower consumed within a building measured over a time period, thebuilding also receiving power produced by a photovoltaic powergeneration plant; a time series of historical measured irradiance dataover the time period; a plurality of photovoltaic plant configurationsthat each comprise a power rating and operational features hypothesizedfor the plant; and a plurality of key parameters comprising one or morecomponent loads, the one or more component loads comprising one of moreof a base load of power consumed, at least one binary load of powerconsumed within the building, and at least one variable load of powerconsumed within the building, wherein the base load represents aconstant power load drawn at all times within the building, the at leastone binary load represents a power load that is either on or off, andwhich, when on, draws power at a single power level within the building,and the at least one variable load represents a power load that can takeon multiple power levels within the building; and a computer comprisinga processor and memory within which code for execution by the processoris stored, the computer configured to: simulate power output productiondata for each of the photovoltaic plant configurations based on anormalized photovoltaic power generation plant using the historicalmeasured irradiance data for the time period; estimate net load dataassociated with one or more of the photovoltaic plant configurations byusing the simulated power output production data for that photovoltaicplant configuration and one or more of the component loads; calculate anerror metric between the time series net load data and the estimated netload data associated with one or more of the photovoltaic plantconfigurations; and infer one of the photovoltaic plant configurationsas a configuration of the photovoltaic plant based on the one or moreerror metrics.
 2. A system according to claim 1, further comprising: anelectric meter configured to measure the time series of net load data.3. A system according to claim 1, wherein the electric meter measuresthe net load data on an hourly basis.
 4. A system according to claim 1,wherein the error metric comprises total squared error and thephotovoltaic plant configuration associated with a minimum one of thetotal squared errors is the inferred configuration of the photovoltaicpower generation plant.
 5. A system according to claim 1, the computeris further configured to: select one or more of the component loads forat least one of the net load estimations to minimize the error metricassociated with that estimated net load.
 6. A system according to claim1, wherein the one or more component loads used during the net loadestimation for at least one of the photovoltaic power generation plantscomprises the binary load that is set as greater than zero only duringspecific hours of a day.
 7. A system according to claim 6, wherein thebinary load comprised in the one or more component loads istemperature-related.
 8. A system according to claim 1, wherein theoperational features of each of the photovoltaic plant configurationsare based on one or more of electrical characteristics, size and numberof photovoltaic arrays, use of fixed or tracking arrays, tilted anglesof elevation, orientation along azimuthal angles, row-to-row shading,shading due to cloud cover, and physical obstructions.
 9. A systemaccording to claim 1, the computer further configured to: obtain a netload data collection comprising the time series of the net load data;and choose the time series of the net load data from the collectionbased on at least one of a power consumption over the time period, aduration of the time period, and an absence of occupants from thebuilding during the time period.
 10. A system according to claim 9,wherein the time period comprises a plurality of contiguous further timeperiods.
 11. A method for net load-based inference of operationalspecifications of a photovoltaic power generation system with the aid ofa digital computer, comprising the steps of: obtaining by a computerwith a time series of net load data for power consumed within a buildingmeasured over a time period, the building also receiving power producedby a photovoltaic power generation plant; obtaining by the computer atime series of historical measured irradiance data over the time period;defining by the computer a plurality of photovoltaic plantconfigurations that each comprise a power rating and operationalfeatures hypothesized for the plant; defining by the computer aplurality of key parameters comprising one or more component loads, theone or more component loads comprising one of more of a base load ofpower consumed, at least one binary load of power consumed within thebuilding, and at least one variable load of power consumed within thebuilding, wherein the base load represents a constant power load drawnat all times within the building, the at least one binary loadrepresents a power load that is either on or off, and which, when on,draws power at a single power level within the building, and the atleast one variable load represents a power load that can take onmultiple power levels within the building; and simulating by thecomputer power output production data for each of the photovoltaic plantconfigurations based on a normalized photovoltaic power generation plantusing the historical measured irradiance data for the time period;estimating by the computer net load data associated with one or more ofthe photovoltaic plant configurations by using the simulated poweroutput production data for that photovoltaic plant configuration and oneor more of the component loads; calculating by the computer an errormetric between the time series net load data and the estimated net loaddata associated with one or more of the photovoltaic plantconfigurations; and inferring by the computer one of the photovoltaicplant configurations as a configuration of the plant based on the one ormore error metrics.
 12. A method according to claim 11, furthercomprising the step of: an electric meter configured to measure the timeseries of net load data.
 13. A method according to claim 11, wherein theelectric meter measures the net load data on an hourly basis.
 14. Amethod according to claim 11, wherein the error metric comprises totalsquared error and the photovoltaic plant configuration associated with aminimum one of the total squared errors is the inferred configuration ofthe photovoltaic power generation plant.
 15. A method according to claim11, further comprising the step of: selecting one or more of thecomponent loads for at least one of the net load estimations to minimizethe error metric associated with that estimated net load.
 16. A methodaccording to claim 11, wherein the one or more component loads usedduring the net load estimation for at least one of the photovoltaicpower generation plants comprises the binary load that is set as greaterthan zero only during specific hours of a day.
 17. A method according toclaim 16, wherein the binary load comprised in the one or more componentloads is temperature-related.
 18. A method according to claim 11,wherein the operational features of each of the photovoltaic plantconfigurations are based on one or more of electrical characteristics,size and number of photovoltaic arrays, use of fixed or tracking arrays,tilted angles of elevation, orientation along azimuthal angles,row-to-row shading, shading due to cloud cover, and physicalobstructions.
 19. A method according to claim 1, further comprising thesteps of: obtain a net load data collection comprising the time seriesof the net load data; and choose the time series of the net load datafrom the collection based on at least one of a power consumption overthe time period, a duration of the time period, and an absence ofoccupants from the building during the time period.
 20. A non-transitorycomputer readable storage medium storing code for executing on acomputer system to perform the method according to claim 11.